LMFDB - Abstract group 1679616.nb: $C_3^8.C_4^2.C_4.C

comment:Define the group as a permutation group

magma:G := PermutationGroup< 36 | (1,12,21,29,2,18,22,36,5,10,20,35,4,13,25,28)(3,15,26,31,8,14,27,34,6,16,24,33,7,17,23,30)(9,11,19,32), (1,15,26,30,2,13,19,36,5,11,23,28,4,10,21,31)(3,14,24,33,8,18,27,32,6,12,25,34,7,17,22,35)(9,16,20,29) >;

gap:G := Group( (1,12,21,29,2,18,22,36,5,10,20,35,4,13,25,28)(3,15,26,31,8,14,27,34,6,16,24,33,7,17,23,30)(9,11,19,32), (1,15,26,30,2,13,19,36,5,11,23,28,4,10,21,31)(3,14,24,33,8,18,27,32,6,12,25,34,7,17,22,35)(9,16,20,29) );

sage:G = PermutationGroup(['(1,12,21,29,2,18,22,36,5,10,20,35,4,13,25,28)(3,15,26,31,8,14,27,34,6,16,24,33,7,17,23,30)(9,11,19,32)', '(1,15,26,30,2,13,19,36,5,11,23,28,4,10,21,31)(3,14,24,33,8,18,27,32,6,12,25,34,7,17,22,35)(9,16,20,29)'])

sage_gap:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(499374196723978425911574054482063250058198293302729003209024818092534050334197520205844896619479935938846404617029536122245712628904436958961647297734333311490026798096660070477068431521795867787030293247936954394799315265688987661399804721923796709360144764707682670524647937733669318904317222630505351824928298013109435811863571061722361359769556165431860338517385692894760441658744858704218596034653643623027269210253864420872365157096529441142652337332389090962893794881565187897742133838853132038547019773786108447994951525319682178274086660676604227770727078486292117426465153098539781195227398399,1679616)'); a = G.1; b = G.4; c = G.7; d = G.10; e = G.11; f = G.12; g = G.13; h = G.14; i = G.15; j = G.16;

Group information

Description:	$C_3^8.C_4^2.C_4.C_2^2$
Order:	$1679616$$\medspace = 2^{8} \cdot 3^{8} $	comment:Order of the group magma:Order(G); gap:Order(G); sage:G.order() sage_gap:G.Order()
Exponent:	$48$$\medspace = 2^{4} \cdot 3 $	comment:Exponent of the group magma:Exponent(G); gap:Exponent(G); sage:G.exponent() sage_gap:G.Exponent()
Automorphism group:	Group of order $26873856$$\medspace = 2^{12} \cdot 3^{8} $	comment:Automorphism group gap:AutomorphismGroup(G); magma:AutomorphismGroup(G); sage_gap:G.AutomorphismGroup()
Composition factors:	$C_2$ x 8, $C_3$ x 8	comment:Composition factors of the group magma:CompositionFactors(G); gap:CompositionSeries(G); sage:G.composition_series() sage_gap:G.CompositionSeries()
Derived length:	$3$	comment:Derived length of the group magma:DerivedLength(G); gap:DerivedLength(G); sage_gap:G.DerivedLength()

This group is nonabelian and solvable. Whether it is monomial or almost simple has not been computed.

comment:Determine if the group G is abelian

magma:IsAbelian(G);

gap:IsAbelian(G);

sage:G.is_abelian()

sage_gap:G.IsAbelian()

comment:Determine if the group G is cyclic

magma:IsCyclic(G);

gap:IsCyclic(G);

sage:G.is_cyclic()

sage_gap:G.IsCyclic()

comment:Determine if the group G is nilpotent

magma:IsNilpotent(G);

gap:IsNilpotentGroup(G);

sage:G.is_nilpotent()

sage_gap:G.IsNilpotentGroup()

comment:Determine if the group G is solvable

magma:IsSolvable(G);

gap:IsSolvableGroup(G);

sage:G.is_solvable()

sage_gap:G.IsSolvableGroup()

comment:Determine if the group G is supersolvable

gap:IsSupersolvableGroup(G);

sage:G.is_supersolvable()

sage_gap:G.IsSupersolvableGroup()

comment:Determine if the group G is simple

magma:IsSimple(G);

gap:IsSimpleGroup(G);

sage_gap:G.IsSimpleGroup()

Group statistics

comment:Compute statistics for the group G

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;

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");

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	6	8	12	16
Elements	1	7047	6560	84888	38880	629856	72576	839808	1679616
Conjugacy classes	1	3	29	8	7	14	8	8	78
Divisions	1	3	29	6	7	7	4	2	59
Autjugacy classes	1	3	10	6	5	8	5	1	39

Minimal presentations

Permutation degree:	$36$
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	32	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Show commands: Gap / Magma / SageMath

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j \mid b^{8}=c^{12}=d^{3}=e^{3}=f^{3}= \!\cdots\! \rangle}$ < a, b, c, d, e, f, g, h, i, j \| b^8,c^12,d^3,e^3,f^3,g^3,h^3,i^3,j^3,[d,e],[d,f],[d,g],[d,h],[d,i],[d,j],[e,f],[e,g],[e,h],[e,i],[e,j],[f,g],[f,h],[f,i],[f,j],[g,h],[g,i],[g,j],[h,i],[h,j],[i,j], b^4d^2e^2f^2ja^-8, b^5c^5dfgh^2i^2ja^-1b^-1a, b^6c^11d^2ef^2i^2a^-1c^-1a, c^8d^2e^2f^2h^2i^2ja^-1d^-1a, c^8df^2g^2h^2ia^-1e^-1a, c^4d^2g^2hi^2ja^-1f^-1a, c^4hi^2ja^-1g^-1a, f^2gi^2a^-1h^-1a, df^2g^2h^2ia^-1i^-1a, c^8g^2h^2a^-1j^-1a, c^9e^2f^2ghb^-1c^-1b, d^2f^2g^2h^2ij^2b^-1d^-1b, fghjb^-1e^-1b, de^2f^2hjb^-1f^-1b, c^8d^2f^2gj^2b^-1g^-1b, c^8d^2egi^2b^-1h^-1b, dei^2b^-1i^-1b, c^8efhij^2b^-1j^-1b, fghi^2jc^-1d^-1c, de^2fghjc^-1e^-1c, de^2g^2h^2j^2c^-1f^-1c, f^2h^2i^2j^2c^-1g^-1c, e^2hc^-1h^-1c, d^2eic^-1i^-1c, fghi^2j^2c^-1j^-1c >
comment:Define the group with the given generators and relations magma:G := PCGroup([16, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 32, 81, 6958498, 69523971, 22120211, 5841059, 179, 113696004, 39036820, 11581796, 228, 40111109, 23557653, 24564517, 43866374, 38001174, 10094374, 1427382, 9193478, 5855670, 326, 52482055, 27484183, 35287079, 25233463, 9622599, 5209687, 375, 147308552, 116527128, 56918056, 16464440, 8234568, 2392, 172482569, 30044185, 59443241, 29752377, 14860873, 7769, 2557545, 2041, 64700426, 74072090, 73869354, 14294074, 14336330, 25434, 1801642, 6458, 199409675, 154877979, 76886059, 15390779, 16923723, 83035, 3873131, 2217723, 207081484, 69941276, 732204, 30065212, 14636620, 269660, 5301612, 2048092, 91349005, 54190109, 80381997, 11153469, 5673549, 871005, 233965, 2059133, 88104974, 156579870, 39121966, 11289662, 5679438, 2799454, 714350, 699966, 42598415, 82346015, 7569455, 44105791, 26185807, 8958047, 6331503, 3168895]); a,b,c,d,e,f,g,h,i,j := Explode([G.1, G.4, G.7, G.10, G.11, G.12, G.13, G.14, G.15, G.16]); AssignNames(~G, ["a", "a2", "a4", "b", "b2", "b4", "c", "c2", "c4", "d", "e", "f", "g", "h", "i", "j"]); gap:G := PcGroupCode(499374196723978425911574054482063250058198293302729003209024818092534050334197520205844896619479935938846404617029536122245712628904436958961647297734333311490026798096660070477068431521795867787030293247936954394799315265688987661399804721923796709360144764707682670524647937733669318904317222630505351824928298013109435811863571061722361359769556165431860338517385692894760441658744858704218596034653643623027269210253864420872365157096529441142652337332389090962893794881565187897742133838853132038547019773786108447994951525319682178274086660676604227770727078486292117426465153098539781195227398399,1679616); a := G.1; b := G.4; c := G.7; d := G.10; e := G.11; f := G.12; g := G.13; h := G.14; i := G.15; j := G.16; sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(499374196723978425911574054482063250058198293302729003209024818092534050334197520205844896619479935938846404617029536122245712628904436958961647297734333311490026798096660070477068431521795867787030293247936954394799315265688987661399804721923796709360144764707682670524647937733669318904317222630505351824928298013109435811863571061722361359769556165431860338517385692894760441658744858704218596034653643623027269210253864420872365157096529441142652337332389090962893794881565187897742133838853132038547019773786108447994951525319682178274086660676604227770727078486292117426465153098539781195227398399,1679616)'); a = G.1; b = G.4; c = G.7; d = G.10; e = G.11; f = G.12; g = G.13; h = G.14; i = G.15; j = G.16; sage_gap:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(499374196723978425911574054482063250058198293302729003209024818092534050334197520205844896619479935938846404617029536122245712628904436958961647297734333311490026798096660070477068431521795867787030293247936954394799315265688987661399804721923796709360144764707682670524647937733669318904317222630505351824928298013109435811863571061722361359769556165431860338517385692894760441658744858704218596034653643623027269210253864420872365157096529441142652337332389090962893794881565187897742133838853132038547019773786108447994951525319682178274086660676604227770727078486292117426465153098539781195227398399,1679616)'); a = G.1; b = G.4; c = G.7; d = G.10; e = G.11; f = G.12; g = G.13; h = G.14; i = G.15; j = G.16;
Permutation group:	Degree $36$ $\langle(1,12,21,29,2,18,22,36,5,10,20,35,4,13,25,28)(3,15,26,31,8,14,27,34,6,16,24,33,7,17,23,30) \!\cdots\! \rangle$ (1,12,21,29,2,18,22,36,5,10,20,35,4,13,25,28)(3,15,26,31,8,14,27,34,6,16,24,33,7,17,23,30)(9,11,19,32), (1,15,26,30,2,13,19,36,5,11,23,28,4,10,21,31)(3,14,24,33,8,18,27,32,6,12,25,34,7,17,22,35)(9,16,20,29)
comment:Define the group as a permutation group magma:G := PermutationGroup< 36 \| (1,12,21,29,2,18,22,36,5,10,20,35,4,13,25,28)(3,15,26,31,8,14,27,34,6,16,24,33,7,17,23,30)(9,11,19,32), (1,15,26,30,2,13,19,36,5,11,23,28,4,10,21,31)(3,14,24,33,8,18,27,32,6,12,25,34,7,17,22,35)(9,16,20,29) >; gap:G := Group( (1,12,21,29,2,18,22,36,5,10,20,35,4,13,25,28)(3,15,26,31,8,14,27,34,6,16,24,33,7,17,23,30)(9,11,19,32), (1,15,26,30,2,13,19,36,5,11,23,28,4,10,21,31)(3,14,24,33,8,18,27,32,6,12,25,34,7,17,22,35)(9,16,20,29) ); sage:G = PermutationGroup(['(1,12,21,29,2,18,22,36,5,10,20,35,4,13,25,28)(3,15,26,31,8,14,27,34,6,16,24,33,7,17,23,30)(9,11,19,32)', '(1,15,26,30,2,13,19,36,5,11,23,28,4,10,21,31)(3,14,24,33,8,18,27,32,6,12,25,34,7,17,22,35)(9,16,20,29)'])
Transitive group:	36T41142	more information
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 permutations of degree 36.

Homology

Abelianization:	$C_{2} \times C_{8} $	comment:The abelianization of the group magma:quo< G \| CommutatorSubgroup(G) >; gap:FactorGroup(G, DerivedSubgroup(G)); sage:G.quotient(G.commutator())
Schur multiplier:	$C_{4}$	comment:The Schur multiplier of the group gap:AbelianInvariantsMultiplier(G); sage:G.homology(2) sage_gap:G.AbelianInvariantsMultiplier()
Commutator length:	$1$	comment:The commutator length of the group gap:CommutatorLength(G); sage_gap:G.CommutatorLength()

Subgroups

comment:List of subgroups of the group

magma:Subgroups(G);

gap:AllSubgroups(G);

sage:G.subgroups()

sage_gap:G.AllSubgroups()

Subgroup data has not been computed.

Character theory

comment:Character table

magma:CharacterTable(G); // Output not guaranteed to exactly match the LMFDB table

gap:CharacterTable(G); # Output not guaranteed to exactly match the LMFDB table

sage:G.character_table() # Output not guaranteed to exactly match the LMFDB table

sage_gap:G.CharacterTable() # Output not guaranteed to exactly match the LMFDB table

Complex character table

The $78 \times 78$ character table is not available for this group.

Rational character table

The $59 \times 59$ rational character table is not available for this group.

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Label	1679616.nb
Order	\( 2^{8} \cdot 3^{8} \)
Exponent	\( 2^{4} \cdot 3 \)

Nilpotent	no
Solvable	yes
$\card{G^{\mathrm{ab}}}$	\( 2^{4} \)
$\card{Z(G)}$	1
$\card{\Aut(G)}$	\( 2^{12} \cdot 3^{8} \)
$\card{\mathrm{Out}(G)}$	\( 2^{4} \)
Perm deg.	$36$
Trans deg.	not computed
Rank	$2$

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