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Properties

Label 167...000.a
Order \( 2^{37} \cdot 3^{18} \cdot 5^{9} \cdot 7^{5} \cdot 11^{3} \cdot 13^{3} \cdot 17^{2} \cdot 19^{2} \cdot 23 \cdot 29 \cdot 31 \cdot 37 \cdot 41 \)
Exponent \( 2^{5} \cdot 3^{3} \cdot 5^{2} \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 37 \cdot 41 \)
Simple yes
$\card{G^{\mathrm{ab}}}$ \( 1 \)
$\card{Z(G)}$ 1
$\card{\Aut(G)}$ \( 2^{38} \cdot 3^{18} \cdot 5^{9} \cdot 7^{5} \cdot 11^{3} \cdot 13^{3} \cdot 17^{2} \cdot 19^{2} \cdot 23 \cdot 29 \cdot 31 \cdot 37 \cdot 41 \)
$\card{\mathrm{Out}(G)}$ \( 2 \)
Perm deg. $41$
Trans deg. not computed
Rank $2$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := AlternatingGroup(41);
 
Copy content gap:G := AlternatingGroup(41);
 
Copy content sage:G = AlternatingGroup(41)
 
Copy content comment:Define the group as a permutation group
 

Group information

Description:$A_{41}$
Order: \(167\!\cdots\!000\)\(\medspace = 2^{37} \cdot 3^{18} \cdot 5^{9} \cdot 7^{5} \cdot 11^{3} \cdot 13^{3} \cdot 17^{2} \cdot 19^{2} \cdot 23 \cdot 29 \cdot 31 \cdot 37 \cdot 41 \)
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: \(219060189739591200\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5^{2} \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 37 \cdot 41 \)
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:Group of order \(334\!\cdots\!000\)\(\medspace = 2^{38} \cdot 3^{18} \cdot 5^{9} \cdot 7^{5} \cdot 11^{3} \cdot 13^{3} \cdot 17^{2} \cdot 19^{2} \cdot 23 \cdot 29 \cdot 31 \cdot 37 \cdot 41 \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$A_{41}$
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:$0$
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 simple (hence nonsolvable, perfect, quasisimple, and almost simple).

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 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 44 45 46 48 50 51 52 54 55 56 57 58 60 62 63 65 66 68 69 70 72 74 75 76 77 78 80 84 85 87 88 90 91 92 93 95 96 99 100 102 104 105 108 110 111 112 114 115 116 117 119 120 124 126 130 132 133 135 136 138 140 143 144 145 150 152 153 154 155 156 160 161 165 168 170 171 174 175 176 180 182 184 186 187 189 190 195 198 200 203 204 207 208 209 210 216 217 220 221 224 225 228 230 231 232 234 238 240 247 248 252 253 255 260 261 264 266 270 272 273 275 276 279 280 285 286 290 297 299 300 304 306 308 310 312 315 319 322 323 325 330 336 340 342 345 348 350 351 357 360 364 368 372 374 378 380 385 390 391 396 399 406 408 414 418 420 429 435 440 442 450 455 456 460 462 465 468 476 483 494 495 504 506 510 520 525 528 532 540 546 550 552 560 561 570 572 580 585 595 598 600 609 612 616 624 627 630 644 646 651 660 663 665 680 684 690 693 700 714 715 720 728 741 748 756 759 760 765 770 780 792 798 805 816 819 825 828 836 840 855 858 870 880 884 897 900 910 912 920 924 935 936 945 952 966 969 975 988 990 1001 1008 1012 1015 1020 1035 1040 1045 1050 1064 1071 1092 1105 1122 1140 1144 1155 1170 1190 1197 1224 1232 1235 1254 1260 1265 1287 1288 1309 1320 1326 1330 1360 1365 1368 1380 1386 1428 1430 1449 1456 1463 1482 1495 1496 1518 1530 1540 1547 1560 1575 1584 1596 1610 1615 1638 1672 1680 1683 1710 1716 1729 1768 1771 1785 1820 1848 1870 1872 1881 1932 1980 1989 1995 2002 2040 2070 2090 2100 2142 2145 2184 2210 2223 2244 2280 2310 2340 2380 2394 2415 2431 2470 2508 2520 2574 2618 2640 2652 2660 2730 2760 2772 2805 2856 2860 2926 2964 3003 3060 3080 3094 3120 3135 3192 3220 3276 3315 3366 3420 3432 3465 3570 3640 3696 3705 3740 3927 3960 3990 4004 4095 4180 4284 4290 4368 4389 4420 4488 4620 4641 4680 4760 4788 5005 5040 5148 5236 5320 5355 5460 5544 5610 5720 5985 6006 6120 6160 6435 6545 6552 6930 7140 7980 8008 8190 8580 9009 9240 10010 10920 12012 13860 15015 16380
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Conjugacy classes   1 10 13 50 8 216 5 82 26 86 3 848 3 46 86 47 2 257 2 265 47 20 1 748 4 15 7 130 1 882 1 5 21 9 21 543 1 6 15 238 2 418 52 78 4 222 9 10 36 14 10 122 8 3 1920 2 45 8 153 20 6 147 337 1 12 15 7 105 80 868 5 4 49 473 6 10 3 4 6 20 13 53 34 118 15 58 2 41 37 3 4 15 4 1052 3 226 40 300 3 8 17 21 258 3 70 2 23 13 9 30 2 195 2 2 51 485 20 7 7 3 15 622 21 7 3 2 4 15 35 76 6 1 90 5 10 2 701 5 1 96 1 1 4 63 8 27 2 48 10 195 1 1 282 1 21 63 3 159 7 6 5 20 3 27 2 157 15 9 2 3 1 18 3 21 47 1 99 59 2 4 1 1 231 87 29 13 10 4 4 2 12 264 30 1 2 4 2 20 11 148 2 90 9 1 40 4 2 894 8 2 55 2 2 8 25 8 107 1 55 14 4 2 26 120 1 62 35 5 26 10 4 67 1 8 30 4 39 11 1 18 5 1 4 1 19 28 15 3 194 4 1 2 274 3 4 14 11 13 15 2 27 4 31 17 3 4 1 2 9 10 38 161 36 18 2 4 9 1 3 3 342 5 20 1 9 3 1 1 24 2 3 120 2 20 1 7 5 1 2 2 57 2 14 1 2 56 3 5 2 2 4 4 71 2 8 31 5 31 31 10 3 6 4 1 5 161 1 3 1 1 96 5 5 1 21 3 7 23 24 7 1 2 1 2 2 2 1 8 37 1 53 2 3 12 1 2 14 1 33 3 5 20 1 1 2 8 22 41 3 1 1 3 42 1 7 3 15 1 1 1 4 7 23 1 2 6 7 72 23 7 1 2 2 1 3 51 4 1 7 3 4 39 1 19 3 6 6 1 1 4 6 14 1 3 2 3 1 9 2 1 2 6 8 11 8 3 2 2 2 12 4 3 5 1 2 10 1 2 1 1 50 2 6 2 1 2 2 2 1 1 3 24 5 2 2 2 3 1 1 3 2 2 7 5 2 1 3 5 2 11 1 5 2 4 1 1 22365
Divisions 1 10 13 50 8 216 5 82 26 86 3 848 3 46 86 47 2 257 2 265 47 20 1 748 4 15 7 130 1 882 1 5 21 9 20 543 1 6 15 238 1 418 52 78 4 222 9 10 36 14 10 122 8 3 1920 2 45 8 153 20 6 147 337 1 11 15 7 105 80 868 5 4 49 473 6 10 3 4 6 20 13 53 34 117 15 58 1 41 37 3 4 15 4 1052 3 226 40 300 3 7 17 21 258 3 70 2 23 13 9 30 2 195 2 2 50 485 20 7 7 3 15 622 21 7 3 2 4 15 35 76 6 1 90 5 10 2 701 5 1 96 1 1 4 63 8 26 2 48 10 195 1 1 282 1 20 63 2 159 7 6 5 19 2 27 1 157 15 9 2 2 1 18 3 21 47 1 99 59 1 4 1 1 231 87 29 13 9 4 4 1 11 264 30 1 2 4 2 20 11 148 1 90 8 1 40 4 2 894 8 2 55 2 2 8 25 8 107 1 55 14 4 2 25 120 1 62 35 4 26 10 4 67 1 8 30 4 39 11 1 17 5 1 4 1 19 28 15 3 194 4 1 1 274 3 4 14 11 13 14 2 27 4 31 17 3 4 1 2 9 9 38 161 36 18 2 4 9 1 3 3 342 5 20 1 9 3 1 1 24 2 3 120 2 20 1 7 5 1 1 2 57 2 14 1 1 56 2 5 2 2 4 4 71 2 8 31 5 29 31 10 3 6 4 1 5 161 1 3 1 1 96 5 5 1 20 3 7 23 24 7 1 2 1 2 1 2 1 8 37 1 53 1 3 12 1 1 14 1 33 2 5 20 1 1 1 8 22 41 3 1 1 3 42 1 6 3 15 1 1 1 4 7 23 1 1 6 7 72 23 7 1 2 1 1 3 51 4 1 7 3 4 39 1 19 3 6 6 1 1 4 6 14 1 3 2 3 1 9 2 1 2 6 8 11 8 3 1 2 2 12 4 3 5 1 2 10 1 1 1 1 50 1 6 2 1 2 2 2 1 1 2 24 5 2 2 1 3 1 1 2 1 2 7 5 2 1 3 5 1 11 1 5 2 4 1 1 22317
Autjugacy classes 1 10 13 50 8 216 5 82 26 86 3 848 3 46 86 47 2 257 2 265 47 20 1 748 4 15 7 130 1 882 1 5 21 9 20 543 1 6 15 238 1 418 52 78 4 222 9 10 36 14 10 122 8 3 1920 2 44 8 153 20 6 147 337 1 11 15 7 105 80 868 5 4 49 473 6 10 3 4 6 20 13 53 34 117 15 58 1 41 37 3 4 15 4 1052 3 226 40 300 3 7 17 21 258 3 70 2 23 13 9 30 2 195 2 2 50 485 20 7 7 3 15 622 21 7 3 2 4 15 35 76 6 1 90 5 10 2 701 5 1 96 1 1 4 63 8 26 2 48 10 195 1 1 282 1 20 63 2 159 7 6 5 19 2 27 1 157 15 9 2 2 1 18 3 21 47 1 99 59 1 4 1 1 231 87 29 13 9 4 4 1 11 264 30 1 2 4 2 20 11 148 1 90 8 1 40 4 2 894 8 2 55 2 2 8 25 8 107 1 55 14 4 2 25 120 1 62 35 4 26 10 4 67 1 8 30 4 39 11 1 17 5 1 4 1 19 28 15 3 194 4 1 1 274 3 4 14 11 13 14 2 27 4 31 17 3 4 1 2 9 9 38 161 36 18 2 4 9 1 3 3 342 5 20 1 9 3 1 1 24 2 3 120 2 20 1 7 5 1 1 2 57 2 14 1 1 56 2 5 2 2 4 4 71 2 8 31 5 29 31 10 3 6 4 1 5 161 1 3 1 1 96 5 5 1 20 3 7 23 24 7 1 2 1 2 1 2 1 8 37 1 53 1 3 12 1 1 14 1 33 2 5 20 1 1 1 8 22 41 3 1 1 3 42 1 6 3 15 1 1 1 4 7 23 1 1 6 7 72 23 7 1 2 1 1 3 51 4 1 7 3 4 39 1 19 3 6 6 1 1 4 6 14 1 3 2 3 1 9 2 1 2 6 8 11 8 3 1 2 2 12 4 3 5 1 2 10 1 1 1 1 50 1 6 2 1 2 2 2 1 1 2 24 5 2 2 1 3 1 1 2 1 2 7 5 2 1 3 5 1 11 1 5 2 4 1 1 22316

Minimal presentations

Permutation degree:$41$
Transitive degree:not computed
Rank: $2$
Inequivalent generating pairs: not computed

Minimal degrees of linear representations for this group have not been computed

Constructions

Show commands: Gap / Magma / SageMath


Permutation group:Degree $41$ $\langle(1,2,3), (3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)\rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 41 | (1,2,3), (3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41) >;
 
Copy content gap:G := Group( (1,2,3), (3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41) );
 
Copy content sage:G = PermutationGroup(['(1,2,3)', '(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)'])
 
Transitive group: 41T9 more information
Direct product: not isomorphic to a non-trivial direct product
Semidirect product: not isomorphic to a non-trivial semidirect product
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as permutations of degree 41.

Homology

Abelianization: $C_1 $
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: not computed
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: not computed
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

The $22365 \times 22365$ character table is not available for this group.

Rational character table

The $22317 \times 22317$ rational character table is not available for this group.