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Properties

Label 26127360.a
Order \( 2^{10} \cdot 3^{6} \cdot 5 \cdot 7 \)
Exponent \( 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Nilpotent no
Solvable no
$\card{G^{\mathrm{ab}}}$ \( 2^{2} \)
$\card{Z(G)}$ 2
$\card{\Aut(G)}$ \( 2^{12} \cdot 3^{6} \cdot 5 \cdot 7 \)
$\card{\mathrm{Out}(G)}$ \( 2^{3} \)
Perm deg. $224$
Trans deg. not computed
Rank $2$

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Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 224 | (1,2)(8,23)(12,18)(14,34)(19,38)(20,42)(21,47)(22,50)(24,54)(25,55)(26,29)(27,30)(28,43)(31,65)(32,70)(33,73)(35,77)(36,78)(37,40)(39,66)(44,93)(45,97)(46,102)(48,105)(49,109)(51,110)(52,113)(53,120)(56,126)(57,122)(58,129)(59,100)(60,115)(61,99)(63,92)(67,141)(68,114)(69,124)(71,147)(72,125)(74,112)(75,151)(76,96)(79,158)(80,157)(81,123)(82,104)(83,106)(84,144)(86,140)(87,169)(89,148)(91,159)(94,127)(95,133)(98,180)(101,185)(103,187)(107,135)(108,193)(111,136)(116,199)(117,200)(118,195)(119,198)(121,203)(128,194)(130,209)(131,183)(132,177)(134,170)(137,172)(142,164)(143,197)(145,176)(146,168)(149,204)(150,166)(152,181)(153,196)(154,179)(155,217)(156,182)(160,201)(161,218)(162,189)(163,174)(165,171)(167,213)(173,212)(175,216)(178,184)(186,211)(188,214)(190,215)(191,221)(192,224)(202,219)(205,222)(206,220)(207,223)(208,210), (3,9)(4,11)(5,15)(6,17)(7,16)(8,24)(10,13)(14,35)(21,48)(22,52)(23,54)(25,57)(32,71)(33,75)(34,77)(36,80)(45,98)(47,105)(49,106)(50,113)(51,61)(53,116)(55,122)(56,117)(58,123)(60,125)(68,143)(70,147)(72,115)(73,151)(74,84)(76,152)(78,157)(79,153)(81,129)(83,109)(87,166)(95,165)(96,181)(97,180)(99,110)(101,173)(108,191)(111,172)(112,144)(114,197)(118,182)(120,199)(121,179)(126,200)(128,205)(132,175)(133,171)(134,142)(136,137)(145,213)(149,207)(150,169)(154,203)(156,195)(158,196)(160,192)(163,214)(164,170)(167,176)(174,188)(177,216)(185,212)(193,221)(194,222)(201,224)(204,223), (1,3,8,22,51,29,11,12,4)(2,5,14,33,74,40,17,18,6)(7,19,41,87,168,178,95,44,20)(9,23,53,119,202,123,54,61,26)(10,27,62,134,189,211,136,63,28)(13,30,64,137,187,215,142,67,31)(15,34,76,155,208,129,77,84,37)(16,38,85,165,183,220,166,86,39)(21,46,101,106,47,104,190,111,49)(24,56,117,52,116,198,210,130,58)(25,59,132,125,55,124,206,133,60)(32,69,145,115,70,100,184,150,72)(35,79,153,75,152,217,219,161,81)(36,82,163,109,78,102,186,164,83)(42,88,171,154,194,222,205,122,89)(43,90,172,156,193,221,191,105,91)(45,96,180,158,200,151,199,143,99)(48,107,188,103,174,92,173,195,108)(50,112,114)(57,127,167,131,176,93,175,203,128)(65,138,170,118,201,224,192,157,135)(66,139,169,121,204,223,207,147,94)(68,120,197,126,196,113,181,98,144)(71,148,216,146,177,140,213,179,149)(73,110,97)(80,159,212,162,185,141,214,182,160), (1,2)(3,7,15,10)(5,13,9,16)(8,21,45,25)(12,18)(14,32,68,36)(19,37,27,29)(20,39,65,43)(22,50)(23,47,97,55)(24,48,98,57)(26,38,40,30)(28,42,66,31)(33,73)(34,70,114,78)(35,71,143,80)(41,62,90,88)(44,92,107,94)(46,100,183,103)(49,84,60,110)(51,109,144,115)(52,113)(53,118,181,121)(54,105,180,122)(56,126)(58,129)(59,131,187,102)(61,83,112,72)(63,135,127,93)(64,85,139,138)(67,140,148,91)(69,104,189,146)(74,125,99,106)(75,151)(76,154,199,156)(77,147,197,157)(79,158)(81,123)(82,162,168,124)(86,89,159,141)(87,167,214,170)(95,111,101,177)(96,179,116,182)(108,192,223,194)(117,200)(119,198)(120,195,152,203)(128,193,224,207)(130,206,219,190)(132,133,136,185)(134,169,213,188)(137,212,175,171)(142,150,145,174)(149,205,221,201)(153,196)(155,217)(160,204,222,191)(161,186,210,184)(163,164,166,176)(165,172,173,216)(178,218,211,208)(202,215,209,220) >;
 
Copy content gap:G := Group( (1,2)(8,23)(12,18)(14,34)(19,38)(20,42)(21,47)(22,50)(24,54)(25,55)(26,29)(27,30)(28,43)(31,65)(32,70)(33,73)(35,77)(36,78)(37,40)(39,66)(44,93)(45,97)(46,102)(48,105)(49,109)(51,110)(52,113)(53,120)(56,126)(57,122)(58,129)(59,100)(60,115)(61,99)(63,92)(67,141)(68,114)(69,124)(71,147)(72,125)(74,112)(75,151)(76,96)(79,158)(80,157)(81,123)(82,104)(83,106)(84,144)(86,140)(87,169)(89,148)(91,159)(94,127)(95,133)(98,180)(101,185)(103,187)(107,135)(108,193)(111,136)(116,199)(117,200)(118,195)(119,198)(121,203)(128,194)(130,209)(131,183)(132,177)(134,170)(137,172)(142,164)(143,197)(145,176)(146,168)(149,204)(150,166)(152,181)(153,196)(154,179)(155,217)(156,182)(160,201)(161,218)(162,189)(163,174)(165,171)(167,213)(173,212)(175,216)(178,184)(186,211)(188,214)(190,215)(191,221)(192,224)(202,219)(205,222)(206,220)(207,223)(208,210), (3,9)(4,11)(5,15)(6,17)(7,16)(8,24)(10,13)(14,35)(21,48)(22,52)(23,54)(25,57)(32,71)(33,75)(34,77)(36,80)(45,98)(47,105)(49,106)(50,113)(51,61)(53,116)(55,122)(56,117)(58,123)(60,125)(68,143)(70,147)(72,115)(73,151)(74,84)(76,152)(78,157)(79,153)(81,129)(83,109)(87,166)(95,165)(96,181)(97,180)(99,110)(101,173)(108,191)(111,172)(112,144)(114,197)(118,182)(120,199)(121,179)(126,200)(128,205)(132,175)(133,171)(134,142)(136,137)(145,213)(149,207)(150,169)(154,203)(156,195)(158,196)(160,192)(163,214)(164,170)(167,176)(174,188)(177,216)(185,212)(193,221)(194,222)(201,224)(204,223), (1,3,8,22,51,29,11,12,4)(2,5,14,33,74,40,17,18,6)(7,19,41,87,168,178,95,44,20)(9,23,53,119,202,123,54,61,26)(10,27,62,134,189,211,136,63,28)(13,30,64,137,187,215,142,67,31)(15,34,76,155,208,129,77,84,37)(16,38,85,165,183,220,166,86,39)(21,46,101,106,47,104,190,111,49)(24,56,117,52,116,198,210,130,58)(25,59,132,125,55,124,206,133,60)(32,69,145,115,70,100,184,150,72)(35,79,153,75,152,217,219,161,81)(36,82,163,109,78,102,186,164,83)(42,88,171,154,194,222,205,122,89)(43,90,172,156,193,221,191,105,91)(45,96,180,158,200,151,199,143,99)(48,107,188,103,174,92,173,195,108)(50,112,114)(57,127,167,131,176,93,175,203,128)(65,138,170,118,201,224,192,157,135)(66,139,169,121,204,223,207,147,94)(68,120,197,126,196,113,181,98,144)(71,148,216,146,177,140,213,179,149)(73,110,97)(80,159,212,162,185,141,214,182,160), (1,2)(3,7,15,10)(5,13,9,16)(8,21,45,25)(12,18)(14,32,68,36)(19,37,27,29)(20,39,65,43)(22,50)(23,47,97,55)(24,48,98,57)(26,38,40,30)(28,42,66,31)(33,73)(34,70,114,78)(35,71,143,80)(41,62,90,88)(44,92,107,94)(46,100,183,103)(49,84,60,110)(51,109,144,115)(52,113)(53,118,181,121)(54,105,180,122)(56,126)(58,129)(59,131,187,102)(61,83,112,72)(63,135,127,93)(64,85,139,138)(67,140,148,91)(69,104,189,146)(74,125,99,106)(75,151)(76,154,199,156)(77,147,197,157)(79,158)(81,123)(82,162,168,124)(86,89,159,141)(87,167,214,170)(95,111,101,177)(96,179,116,182)(108,192,223,194)(117,200)(119,198)(120,195,152,203)(128,193,224,207)(130,206,219,190)(132,133,136,185)(134,169,213,188)(137,212,175,171)(142,150,145,174)(149,205,221,201)(153,196)(155,217)(160,204,222,191)(161,186,210,184)(163,164,166,176)(165,172,173,216)(178,218,211,208)(202,215,209,220) );
 
Copy content sage:G = PermutationGroup(['(1,2)(8,23)(12,18)(14,34)(19,38)(20,42)(21,47)(22,50)(24,54)(25,55)(26,29)(27,30)(28,43)(31,65)(32,70)(33,73)(35,77)(36,78)(37,40)(39,66)(44,93)(45,97)(46,102)(48,105)(49,109)(51,110)(52,113)(53,120)(56,126)(57,122)(58,129)(59,100)(60,115)(61,99)(63,92)(67,141)(68,114)(69,124)(71,147)(72,125)(74,112)(75,151)(76,96)(79,158)(80,157)(81,123)(82,104)(83,106)(84,144)(86,140)(87,169)(89,148)(91,159)(94,127)(95,133)(98,180)(101,185)(103,187)(107,135)(108,193)(111,136)(116,199)(117,200)(118,195)(119,198)(121,203)(128,194)(130,209)(131,183)(132,177)(134,170)(137,172)(142,164)(143,197)(145,176)(146,168)(149,204)(150,166)(152,181)(153,196)(154,179)(155,217)(156,182)(160,201)(161,218)(162,189)(163,174)(165,171)(167,213)(173,212)(175,216)(178,184)(186,211)(188,214)(190,215)(191,221)(192,224)(202,219)(205,222)(206,220)(207,223)(208,210)', '(3,9)(4,11)(5,15)(6,17)(7,16)(8,24)(10,13)(14,35)(21,48)(22,52)(23,54)(25,57)(32,71)(33,75)(34,77)(36,80)(45,98)(47,105)(49,106)(50,113)(51,61)(53,116)(55,122)(56,117)(58,123)(60,125)(68,143)(70,147)(72,115)(73,151)(74,84)(76,152)(78,157)(79,153)(81,129)(83,109)(87,166)(95,165)(96,181)(97,180)(99,110)(101,173)(108,191)(111,172)(112,144)(114,197)(118,182)(120,199)(121,179)(126,200)(128,205)(132,175)(133,171)(134,142)(136,137)(145,213)(149,207)(150,169)(154,203)(156,195)(158,196)(160,192)(163,214)(164,170)(167,176)(174,188)(177,216)(185,212)(193,221)(194,222)(201,224)(204,223)', '(1,3,8,22,51,29,11,12,4)(2,5,14,33,74,40,17,18,6)(7,19,41,87,168,178,95,44,20)(9,23,53,119,202,123,54,61,26)(10,27,62,134,189,211,136,63,28)(13,30,64,137,187,215,142,67,31)(15,34,76,155,208,129,77,84,37)(16,38,85,165,183,220,166,86,39)(21,46,101,106,47,104,190,111,49)(24,56,117,52,116,198,210,130,58)(25,59,132,125,55,124,206,133,60)(32,69,145,115,70,100,184,150,72)(35,79,153,75,152,217,219,161,81)(36,82,163,109,78,102,186,164,83)(42,88,171,154,194,222,205,122,89)(43,90,172,156,193,221,191,105,91)(45,96,180,158,200,151,199,143,99)(48,107,188,103,174,92,173,195,108)(50,112,114)(57,127,167,131,176,93,175,203,128)(65,138,170,118,201,224,192,157,135)(66,139,169,121,204,223,207,147,94)(68,120,197,126,196,113,181,98,144)(71,148,216,146,177,140,213,179,149)(73,110,97)(80,159,212,162,185,141,214,182,160)', '(1,2)(3,7,15,10)(5,13,9,16)(8,21,45,25)(12,18)(14,32,68,36)(19,37,27,29)(20,39,65,43)(22,50)(23,47,97,55)(24,48,98,57)(26,38,40,30)(28,42,66,31)(33,73)(34,70,114,78)(35,71,143,80)(41,62,90,88)(44,92,107,94)(46,100,183,103)(49,84,60,110)(51,109,144,115)(52,113)(53,118,181,121)(54,105,180,122)(56,126)(58,129)(59,131,187,102)(61,83,112,72)(63,135,127,93)(64,85,139,138)(67,140,148,91)(69,104,189,146)(74,125,99,106)(75,151)(76,154,199,156)(77,147,197,157)(79,158)(81,123)(82,162,168,124)(86,89,159,141)(87,167,214,170)(95,111,101,177)(96,179,116,182)(108,192,223,194)(117,200)(119,198)(120,195,152,203)(128,193,224,207)(130,206,219,190)(132,133,136,185)(134,169,213,188)(137,212,175,171)(142,150,145,174)(149,205,221,201)(153,196)(155,217)(160,204,222,191)(161,186,210,184)(163,164,166,176)(165,172,173,216)(178,218,211,208)(202,215,209,220)'])
 

Group information

Description:$\GOrthMinus(6,3)$
Order: \(26127360\)\(\medspace = 2^{10} \cdot 3^{6} \cdot 5 \cdot 7 \)
Copy content comment:Order of the group
 
Copy content magma:Order(G);
 
Copy content gap:Order(G);
 
Copy content sage:G.order()
 
Copy content sage_gap:G.Order()
 
Exponent: \(2520\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Copy content comment:Exponent of the group
 
Copy content magma:Exponent(G);
 
Copy content gap:Exponent(G);
 
Copy content sage:G.exponent()
 
Copy content sage_gap:G.Exponent()
 
Automorphism group:$C_2^2.\PSU(4,3).D_4$, of order \(104509440\)\(\medspace = 2^{12} \cdot 3^{6} \cdot 5 \cdot 7 \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$C_2$ x 3, $\PSU(4,3)$
Copy content comment:Composition factors of the group
 
Copy content magma:CompositionFactors(G);
 
Copy content gap:CompositionSeries(G);
 
Copy content sage:G.composition_series()
 
Copy content sage_gap:G.CompositionSeries()
 
Derived length:$1$
Copy content comment:Derived length of the group
 
Copy content magma:DerivedLength(G);
 
Copy content gap:DerivedLength(G);
 
Copy content sage_gap:G.DerivedLength()
 

This group is nonabelian and nonsolvable. Whether it is almost simple has not been computed.

Copy content comment:Determine if the group G is abelian
 
Copy content magma:IsAbelian(G);
 
Copy content gap:IsAbelian(G);
 
Copy content sage:G.is_abelian()
 
Copy content sage_gap:G.IsAbelian()
 
Copy content comment:Determine if the group G is cyclic
 
Copy content magma:IsCyclic(G);
 
Copy content gap:IsCyclic(G);
 
Copy content sage:G.is_cyclic()
 
Copy content sage_gap:G.IsCyclic()
 
Copy content comment:Determine if the group G is nilpotent
 
Copy content magma:IsNilpotent(G);
 
Copy content gap:IsNilpotentGroup(G);
 
Copy content sage:G.is_nilpotent()
 
Copy content sage_gap:G.IsNilpotentGroup()
 
Copy content comment:Determine if the group G is solvable
 
Copy content magma:IsSolvable(G);
 
Copy content gap:IsSolvableGroup(G);
 
Copy content sage:G.is_solvable()
 
Copy content sage_gap:G.IsSolvableGroup()
 
Copy content comment:Determine if the group G is supersolvable
 
Copy content gap:IsSupersolvableGroup(G);
 
Copy content sage:G.is_supersolvable()
 
Copy content sage_gap:G.IsSupersolvableGroup()
 
Copy content comment:Determine if the group G is simple
 
Copy content magma:IsSimple(G);
 
Copy content gap:IsSimpleGroup(G);
 
Copy content sage_gap:G.IsSimpleGroup()
 

Group statistics

Copy content comment:Compute statistics for the group G
 
Copy content magma:// Magma code to output the first two rows of the group statistics table element_orders := [Order(g) : g in G]; orders := Set(element_orders); printf "Orders: %o\n", orders; printf "Elements: %o %o\n", [#[x : x in element_orders | x eq n] : n in orders], Order(G); cc_orders := [cc[1] : cc in ConjugacyClasses(G)]; printf "Conjugacy classes: %o %o\n", [#[x : x in cc_orders | x eq n] : n in orders], #cc_orders;
 
Copy content gap:# Gap code to output the first two rows of the group statistics table element_orders := List(Elements(G), g -> Order(g)); orders := Set(element_orders); Print("Orders: ", orders, "\n"); element_counts := List(orders, n -> Length(Filtered(element_orders, x -> x = n))); Print("Elements: ", element_counts, " ", Size(G), "\n"); cc_orders := List(ConjugacyClasses(G), cc -> Order(Representative(cc))); cc_counts := List(orders, n -> Length(Filtered(cc_orders, x -> x = n))); Print("Conjugacy classes: ", cc_counts, " ", Length(ConjugacyClasses(G)), "\n");
 
Copy content sage:# Sage code to output the first two rows of the group statistics table element_orders = [g.order() for g in G] orders = sorted(list(set(element_orders))) print("Orders:", orders) print("Elements:", [element_orders.count(n) for n in orders], G.order()) cc_orders = [cc[0].order() for cc in G.conjugacy_classes()] print("Conjugacy classes:", [cc_orders.count(n) for n in orders], len(cc_orders))
 

Order 1 2 3 4 5 6 7 8 9 10 12 14 18 28
Elements 1 37927 47600 999000 653184 3444560 933120 3265920 483840 4572288 5503680 933120 3386880 1866240 26127360
Conjugacy classes   1 11 4 13 1 32 1 5 2 7 16 1 6 2 102
Divisions 1 11 4 13 1 32 1 5 2 7 16 1 6 1 101
Autjugacy classes 1 6 3 8 1 13 1 3 1 3 8 1 2 1 52

Copy content comment:Compute statistics about the characters of G
 
Copy content magma:// Outputs [<d_1,c_1>, <d_2,c_2>, ...] where c_i is the number of irr. complex chars. of G with degree d_i CharacterDegrees(G);
 
Copy content gap:# Outputs [[d_1,c_1], [d_2,c_2], ...] where c_i is the number of irr. complex chars. of G with degree d_i CharacterDegrees(G);
 
Copy content sage:# Outputs [[d_1,c_1], [d_2,c_2], ...] where c_i is the number of irr. complex chars. of G with degree d_i character_degrees = [c[0] for c in G.character_table()] [[n, character_degrees.count(n)] for n in set(character_degrees)]
 
Copy content sage_gap:G.CharacterDegrees()
 

Dimension 1 21 35 40 56 70 90 140 189 210 240 315 420 504 560 630 729 896 1080 1120 1280 2560
Irr. complex chars.   4 4 8 1 8 4 4 8 4 4 1 8 6 8 8 4 4 8 1 1 4 0 102
Irr. rational chars. 4 4 8 1 8 4 4 8 4 4 1 8 6 8 8 4 4 8 1 1 2 1 101

Minimal presentations

Permutation degree:$224$
Transitive degree:not computed
Rank: $2$
Inequivalent generating pairs: not computed

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 40 40 40
Arbitrary not computed not computed not computed

Constructions

Show commands: Gap / Magma / SageMath


Groups of Lie type:$\GOMinus(6,3)$
Permutation group:Degree $224$ $\langle(1,2)(8,23)(12,18)(14,34)(19,38)(20,42)(21,47)(22,50)(24,54)(25,55)(26,29) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 224 | (1,2)(8,23)(12,18)(14,34)(19,38)(20,42)(21,47)(22,50)(24,54)(25,55)(26,29)(27,30)(28,43)(31,65)(32,70)(33,73)(35,77)(36,78)(37,40)(39,66)(44,93)(45,97)(46,102)(48,105)(49,109)(51,110)(52,113)(53,120)(56,126)(57,122)(58,129)(59,100)(60,115)(61,99)(63,92)(67,141)(68,114)(69,124)(71,147)(72,125)(74,112)(75,151)(76,96)(79,158)(80,157)(81,123)(82,104)(83,106)(84,144)(86,140)(87,169)(89,148)(91,159)(94,127)(95,133)(98,180)(101,185)(103,187)(107,135)(108,193)(111,136)(116,199)(117,200)(118,195)(119,198)(121,203)(128,194)(130,209)(131,183)(132,177)(134,170)(137,172)(142,164)(143,197)(145,176)(146,168)(149,204)(150,166)(152,181)(153,196)(154,179)(155,217)(156,182)(160,201)(161,218)(162,189)(163,174)(165,171)(167,213)(173,212)(175,216)(178,184)(186,211)(188,214)(190,215)(191,221)(192,224)(202,219)(205,222)(206,220)(207,223)(208,210), (3,9)(4,11)(5,15)(6,17)(7,16)(8,24)(10,13)(14,35)(21,48)(22,52)(23,54)(25,57)(32,71)(33,75)(34,77)(36,80)(45,98)(47,105)(49,106)(50,113)(51,61)(53,116)(55,122)(56,117)(58,123)(60,125)(68,143)(70,147)(72,115)(73,151)(74,84)(76,152)(78,157)(79,153)(81,129)(83,109)(87,166)(95,165)(96,181)(97,180)(99,110)(101,173)(108,191)(111,172)(112,144)(114,197)(118,182)(120,199)(121,179)(126,200)(128,205)(132,175)(133,171)(134,142)(136,137)(145,213)(149,207)(150,169)(154,203)(156,195)(158,196)(160,192)(163,214)(164,170)(167,176)(174,188)(177,216)(185,212)(193,221)(194,222)(201,224)(204,223), (1,3,8,22,51,29,11,12,4)(2,5,14,33,74,40,17,18,6)(7,19,41,87,168,178,95,44,20)(9,23,53,119,202,123,54,61,26)(10,27,62,134,189,211,136,63,28)(13,30,64,137,187,215,142,67,31)(15,34,76,155,208,129,77,84,37)(16,38,85,165,183,220,166,86,39)(21,46,101,106,47,104,190,111,49)(24,56,117,52,116,198,210,130,58)(25,59,132,125,55,124,206,133,60)(32,69,145,115,70,100,184,150,72)(35,79,153,75,152,217,219,161,81)(36,82,163,109,78,102,186,164,83)(42,88,171,154,194,222,205,122,89)(43,90,172,156,193,221,191,105,91)(45,96,180,158,200,151,199,143,99)(48,107,188,103,174,92,173,195,108)(50,112,114)(57,127,167,131,176,93,175,203,128)(65,138,170,118,201,224,192,157,135)(66,139,169,121,204,223,207,147,94)(68,120,197,126,196,113,181,98,144)(71,148,216,146,177,140,213,179,149)(73,110,97)(80,159,212,162,185,141,214,182,160), (1,2)(3,7,15,10)(5,13,9,16)(8,21,45,25)(12,18)(14,32,68,36)(19,37,27,29)(20,39,65,43)(22,50)(23,47,97,55)(24,48,98,57)(26,38,40,30)(28,42,66,31)(33,73)(34,70,114,78)(35,71,143,80)(41,62,90,88)(44,92,107,94)(46,100,183,103)(49,84,60,110)(51,109,144,115)(52,113)(53,118,181,121)(54,105,180,122)(56,126)(58,129)(59,131,187,102)(61,83,112,72)(63,135,127,93)(64,85,139,138)(67,140,148,91)(69,104,189,146)(74,125,99,106)(75,151)(76,154,199,156)(77,147,197,157)(79,158)(81,123)(82,162,168,124)(86,89,159,141)(87,167,214,170)(95,111,101,177)(96,179,116,182)(108,192,223,194)(117,200)(119,198)(120,195,152,203)(128,193,224,207)(130,206,219,190)(132,133,136,185)(134,169,213,188)(137,212,175,171)(142,150,145,174)(149,205,221,201)(153,196)(155,217)(160,204,222,191)(161,186,210,184)(163,164,166,176)(165,172,173,216)(178,218,211,208)(202,215,209,220) >;
 
Copy content gap:G := Group( (1,2)(8,23)(12,18)(14,34)(19,38)(20,42)(21,47)(22,50)(24,54)(25,55)(26,29)(27,30)(28,43)(31,65)(32,70)(33,73)(35,77)(36,78)(37,40)(39,66)(44,93)(45,97)(46,102)(48,105)(49,109)(51,110)(52,113)(53,120)(56,126)(57,122)(58,129)(59,100)(60,115)(61,99)(63,92)(67,141)(68,114)(69,124)(71,147)(72,125)(74,112)(75,151)(76,96)(79,158)(80,157)(81,123)(82,104)(83,106)(84,144)(86,140)(87,169)(89,148)(91,159)(94,127)(95,133)(98,180)(101,185)(103,187)(107,135)(108,193)(111,136)(116,199)(117,200)(118,195)(119,198)(121,203)(128,194)(130,209)(131,183)(132,177)(134,170)(137,172)(142,164)(143,197)(145,176)(146,168)(149,204)(150,166)(152,181)(153,196)(154,179)(155,217)(156,182)(160,201)(161,218)(162,189)(163,174)(165,171)(167,213)(173,212)(175,216)(178,184)(186,211)(188,214)(190,215)(191,221)(192,224)(202,219)(205,222)(206,220)(207,223)(208,210), (3,9)(4,11)(5,15)(6,17)(7,16)(8,24)(10,13)(14,35)(21,48)(22,52)(23,54)(25,57)(32,71)(33,75)(34,77)(36,80)(45,98)(47,105)(49,106)(50,113)(51,61)(53,116)(55,122)(56,117)(58,123)(60,125)(68,143)(70,147)(72,115)(73,151)(74,84)(76,152)(78,157)(79,153)(81,129)(83,109)(87,166)(95,165)(96,181)(97,180)(99,110)(101,173)(108,191)(111,172)(112,144)(114,197)(118,182)(120,199)(121,179)(126,200)(128,205)(132,175)(133,171)(134,142)(136,137)(145,213)(149,207)(150,169)(154,203)(156,195)(158,196)(160,192)(163,214)(164,170)(167,176)(174,188)(177,216)(185,212)(193,221)(194,222)(201,224)(204,223), (1,3,8,22,51,29,11,12,4)(2,5,14,33,74,40,17,18,6)(7,19,41,87,168,178,95,44,20)(9,23,53,119,202,123,54,61,26)(10,27,62,134,189,211,136,63,28)(13,30,64,137,187,215,142,67,31)(15,34,76,155,208,129,77,84,37)(16,38,85,165,183,220,166,86,39)(21,46,101,106,47,104,190,111,49)(24,56,117,52,116,198,210,130,58)(25,59,132,125,55,124,206,133,60)(32,69,145,115,70,100,184,150,72)(35,79,153,75,152,217,219,161,81)(36,82,163,109,78,102,186,164,83)(42,88,171,154,194,222,205,122,89)(43,90,172,156,193,221,191,105,91)(45,96,180,158,200,151,199,143,99)(48,107,188,103,174,92,173,195,108)(50,112,114)(57,127,167,131,176,93,175,203,128)(65,138,170,118,201,224,192,157,135)(66,139,169,121,204,223,207,147,94)(68,120,197,126,196,113,181,98,144)(71,148,216,146,177,140,213,179,149)(73,110,97)(80,159,212,162,185,141,214,182,160), (1,2)(3,7,15,10)(5,13,9,16)(8,21,45,25)(12,18)(14,32,68,36)(19,37,27,29)(20,39,65,43)(22,50)(23,47,97,55)(24,48,98,57)(26,38,40,30)(28,42,66,31)(33,73)(34,70,114,78)(35,71,143,80)(41,62,90,88)(44,92,107,94)(46,100,183,103)(49,84,60,110)(51,109,144,115)(52,113)(53,118,181,121)(54,105,180,122)(56,126)(58,129)(59,131,187,102)(61,83,112,72)(63,135,127,93)(64,85,139,138)(67,140,148,91)(69,104,189,146)(74,125,99,106)(75,151)(76,154,199,156)(77,147,197,157)(79,158)(81,123)(82,162,168,124)(86,89,159,141)(87,167,214,170)(95,111,101,177)(96,179,116,182)(108,192,223,194)(117,200)(119,198)(120,195,152,203)(128,193,224,207)(130,206,219,190)(132,133,136,185)(134,169,213,188)(137,212,175,171)(142,150,145,174)(149,205,221,201)(153,196)(155,217)(160,204,222,191)(161,186,210,184)(163,164,166,176)(165,172,173,216)(178,218,211,208)(202,215,209,220) );
 
Copy content sage:G = PermutationGroup(['(1,2)(8,23)(12,18)(14,34)(19,38)(20,42)(21,47)(22,50)(24,54)(25,55)(26,29)(27,30)(28,43)(31,65)(32,70)(33,73)(35,77)(36,78)(37,40)(39,66)(44,93)(45,97)(46,102)(48,105)(49,109)(51,110)(52,113)(53,120)(56,126)(57,122)(58,129)(59,100)(60,115)(61,99)(63,92)(67,141)(68,114)(69,124)(71,147)(72,125)(74,112)(75,151)(76,96)(79,158)(80,157)(81,123)(82,104)(83,106)(84,144)(86,140)(87,169)(89,148)(91,159)(94,127)(95,133)(98,180)(101,185)(103,187)(107,135)(108,193)(111,136)(116,199)(117,200)(118,195)(119,198)(121,203)(128,194)(130,209)(131,183)(132,177)(134,170)(137,172)(142,164)(143,197)(145,176)(146,168)(149,204)(150,166)(152,181)(153,196)(154,179)(155,217)(156,182)(160,201)(161,218)(162,189)(163,174)(165,171)(167,213)(173,212)(175,216)(178,184)(186,211)(188,214)(190,215)(191,221)(192,224)(202,219)(205,222)(206,220)(207,223)(208,210)', '(3,9)(4,11)(5,15)(6,17)(7,16)(8,24)(10,13)(14,35)(21,48)(22,52)(23,54)(25,57)(32,71)(33,75)(34,77)(36,80)(45,98)(47,105)(49,106)(50,113)(51,61)(53,116)(55,122)(56,117)(58,123)(60,125)(68,143)(70,147)(72,115)(73,151)(74,84)(76,152)(78,157)(79,153)(81,129)(83,109)(87,166)(95,165)(96,181)(97,180)(99,110)(101,173)(108,191)(111,172)(112,144)(114,197)(118,182)(120,199)(121,179)(126,200)(128,205)(132,175)(133,171)(134,142)(136,137)(145,213)(149,207)(150,169)(154,203)(156,195)(158,196)(160,192)(163,214)(164,170)(167,176)(174,188)(177,216)(185,212)(193,221)(194,222)(201,224)(204,223)', '(1,3,8,22,51,29,11,12,4)(2,5,14,33,74,40,17,18,6)(7,19,41,87,168,178,95,44,20)(9,23,53,119,202,123,54,61,26)(10,27,62,134,189,211,136,63,28)(13,30,64,137,187,215,142,67,31)(15,34,76,155,208,129,77,84,37)(16,38,85,165,183,220,166,86,39)(21,46,101,106,47,104,190,111,49)(24,56,117,52,116,198,210,130,58)(25,59,132,125,55,124,206,133,60)(32,69,145,115,70,100,184,150,72)(35,79,153,75,152,217,219,161,81)(36,82,163,109,78,102,186,164,83)(42,88,171,154,194,222,205,122,89)(43,90,172,156,193,221,191,105,91)(45,96,180,158,200,151,199,143,99)(48,107,188,103,174,92,173,195,108)(50,112,114)(57,127,167,131,176,93,175,203,128)(65,138,170,118,201,224,192,157,135)(66,139,169,121,204,223,207,147,94)(68,120,197,126,196,113,181,98,144)(71,148,216,146,177,140,213,179,149)(73,110,97)(80,159,212,162,185,141,214,182,160)', '(1,2)(3,7,15,10)(5,13,9,16)(8,21,45,25)(12,18)(14,32,68,36)(19,37,27,29)(20,39,65,43)(22,50)(23,47,97,55)(24,48,98,57)(26,38,40,30)(28,42,66,31)(33,73)(34,70,114,78)(35,71,143,80)(41,62,90,88)(44,92,107,94)(46,100,183,103)(49,84,60,110)(51,109,144,115)(52,113)(53,118,181,121)(54,105,180,122)(56,126)(58,129)(59,131,187,102)(61,83,112,72)(63,135,127,93)(64,85,139,138)(67,140,148,91)(69,104,189,146)(74,125,99,106)(75,151)(76,154,199,156)(77,147,197,157)(79,158)(81,123)(82,162,168,124)(86,89,159,141)(87,167,214,170)(95,111,101,177)(96,179,116,182)(108,192,223,194)(117,200)(119,198)(120,195,152,203)(128,193,224,207)(130,206,219,190)(132,133,136,185)(134,169,213,188)(137,212,175,171)(142,150,145,174)(149,205,221,201)(153,196)(155,217)(160,204,222,191)(161,186,210,184)(163,164,166,176)(165,172,173,216)(178,218,211,208)(202,215,209,220)'])
 
Direct product: not computed
Semidirect product: not computed
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as matrices in $\GOMinus(6,3)$.

Homology

Abelianization: $C_{2}^{2} $
Copy content comment:The abelianization of the group
 
Copy content magma:quo< G | CommutatorSubgroup(G) >;
 
Copy content gap:FactorGroup(G, DerivedSubgroup(G));
 
Copy content sage:G.quotient(G.commutator())
 
Schur multiplier: $C_{2}^{2}$
Copy content comment:The Schur multiplier of the group
 
Copy content gap:AbelianInvariantsMultiplier(G);
 
Copy content sage:G.homology(2)
 
Copy content sage_gap:G.AbelianInvariantsMultiplier()
 
Commutator length: $1$
Copy content comment:The commutator length of the group
 
Copy content gap:CommutatorLength(G);
 
Copy content sage_gap:G.CommutatorLength()
 

Subgroups

Copy content comment:List of subgroups of the group
 
Copy content magma:Subgroups(G);
 
Copy content gap:AllSubgroups(G);
 
Copy content sage:G.subgroups()
 
Copy content sage_gap:G.AllSubgroups()
 

Subgroup data has not been computed.

Character theory

Copy content comment:Character table
 
Copy content magma:CharacterTable(G); // Output not guaranteed to exactly match the LMFDB table
 
Copy content gap:CharacterTable(G); # Output not guaranteed to exactly match the LMFDB table
 
Copy content sage:G.character_table() # Output not guaranteed to exactly match the LMFDB table
 
Copy content sage_gap:G.CharacterTable() # Output not guaranteed to exactly match the LMFDB table
 

Complex character table

See the $102 \times 102$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $101 \times 101$ rational character table.