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G = C3×D8⋊S3order 288 = 25·32

Direct product of C3 and D8⋊S3

direct product, metabelian, supersoluble, monomial

Aliases: C3×D8⋊S3, C2412D6, C82(S3×C6), C244(C2×C6), D4⋊S32C6, D42(S3×C6), D82(C3×S3), (C3×D8)⋊6S3, (S3×D4)⋊2C6, (C3×D8)⋊4C6, C8⋊S33C6, C24⋊C23C6, (C3×D4)⋊13D6, D4.S31C6, D6.6(C3×D4), C6.28(C6×D4), D42S31C6, Dic61(C2×C6), (S3×C6).42D4, D12.1(C2×C6), C6.188(S3×D4), (C32×D8)⋊8C2, (C3×C24)⋊13C22, C12.2(C22×C6), Dic3.8(C3×D4), C3218(C8⋊C22), (C3×C12).73C23, (C3×Dic3).45D4, (D4×C32)⋊6C22, (S3×C12).26C22, C12.153(C22×S3), (C3×Dic6)⋊11C22, (C3×D12).25C22, C3⋊C81(C2×C6), (C3×S3×D4)⋊5C2, C4.2(S3×C2×C6), C2.16(C3×S3×D4), (C3×D4)⋊2(C2×C6), C32(C3×C8⋊C22), (C3×C8⋊S3)⋊7C2, (C3×C24⋊C2)⋊7C2, (C3×D4⋊S3)⋊11C2, (C3×C3⋊C8)⋊18C22, (C4×S3).1(C2×C6), (C3×D4.S3)⋊9C2, (C3×D42S3)⋊4C2, (C3×C6).216(C2×D4), SmallGroup(288,682)

Series: Derived Chief Lower central Upper central

C1C12 — C3×D8⋊S3
C1C3C6C12C3×C12S3×C12C3×S3×D4 — C3×D8⋊S3
C3C6C12 — C3×D8⋊S3
C1C6C12C3×D8

Generators and relations for C3×D8⋊S3
 G = < a,b,c,d,e | a3=b8=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b5, cd=dc, ce=ec, ede=d-1 >

Subgroups: 426 in 147 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, M4(2), D8, D8, SD16, C2×D4, C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×C6, C8⋊C22, C3×Dic3, C3×Dic3, C3×C12, S3×C6, S3×C6, C62, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×M4(2), C3×D8, C3×D8, C3×SD16, S3×D4, D42S3, C6×D4, C3×C4○D4, C3×C3⋊C8, C3×C24, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, D4×C32, S3×C2×C6, D8⋊S3, C3×C8⋊C22, C3×C8⋊S3, C3×C24⋊C2, C3×D4⋊S3, C3×D4.S3, C32×D8, C3×S3×D4, C3×D42S3, C3×D8⋊S3
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, C3×D4, C22×S3, C22×C6, C8⋊C22, S3×C6, S3×D4, C6×D4, S3×C2×C6, D8⋊S3, C3×C8⋊C22, C3×S3×D4, C3×D8⋊S3

Smallest permutation representation of C3×D8⋊S3
On 48 points
Generators in S48
(1 37 32)(2 38 25)(3 39 26)(4 40 27)(5 33 28)(6 34 29)(7 35 30)(8 36 31)(9 21 41)(10 22 42)(11 23 43)(12 24 44)(13 17 45)(14 18 46)(15 19 47)(16 20 48)
(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 43 44 45 46 47 48)
(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)(17 19)(20 24)(21 23)(25 31)(26 30)(27 29)(34 40)(35 39)(36 38)(41 43)(44 48)(45 47)
(1 32 37)(2 25 38)(3 26 39)(4 27 40)(5 28 33)(6 29 34)(7 30 35)(8 31 36)(9 21 41)(10 22 42)(11 23 43)(12 24 44)(13 17 45)(14 18 46)(15 19 47)(16 20 48)
(1 18)(2 23)(3 20)(4 17)(5 22)(6 19)(7 24)(8 21)(9 31)(10 28)(11 25)(12 30)(13 27)(14 32)(15 29)(16 26)(33 42)(34 47)(35 44)(36 41)(37 46)(38 43)(39 48)(40 45)

G:=sub<Sym(48)| (1,37,32)(2,38,25)(3,39,26)(4,40,27)(5,33,28)(6,34,29)(7,35,30)(8,36,31)(9,21,41)(10,22,42)(11,23,43)(12,24,44)(13,17,45)(14,18,46)(15,19,47)(16,20,48), (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)(41,42,43,44,45,46,47,48), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)(17,19)(20,24)(21,23)(25,31)(26,30)(27,29)(34,40)(35,39)(36,38)(41,43)(44,48)(45,47), (1,32,37)(2,25,38)(3,26,39)(4,27,40)(5,28,33)(6,29,34)(7,30,35)(8,31,36)(9,21,41)(10,22,42)(11,23,43)(12,24,44)(13,17,45)(14,18,46)(15,19,47)(16,20,48), (1,18)(2,23)(3,20)(4,17)(5,22)(6,19)(7,24)(8,21)(9,31)(10,28)(11,25)(12,30)(13,27)(14,32)(15,29)(16,26)(33,42)(34,47)(35,44)(36,41)(37,46)(38,43)(39,48)(40,45)>;

G:=Group( (1,37,32)(2,38,25)(3,39,26)(4,40,27)(5,33,28)(6,34,29)(7,35,30)(8,36,31)(9,21,41)(10,22,42)(11,23,43)(12,24,44)(13,17,45)(14,18,46)(15,19,47)(16,20,48), (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)(41,42,43,44,45,46,47,48), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)(17,19)(20,24)(21,23)(25,31)(26,30)(27,29)(34,40)(35,39)(36,38)(41,43)(44,48)(45,47), (1,32,37)(2,25,38)(3,26,39)(4,27,40)(5,28,33)(6,29,34)(7,30,35)(8,31,36)(9,21,41)(10,22,42)(11,23,43)(12,24,44)(13,17,45)(14,18,46)(15,19,47)(16,20,48), (1,18)(2,23)(3,20)(4,17)(5,22)(6,19)(7,24)(8,21)(9,31)(10,28)(11,25)(12,30)(13,27)(14,32)(15,29)(16,26)(33,42)(34,47)(35,44)(36,41)(37,46)(38,43)(39,48)(40,45) );

G=PermutationGroup([[(1,37,32),(2,38,25),(3,39,26),(4,40,27),(5,33,28),(6,34,29),(7,35,30),(8,36,31),(9,21,41),(10,22,42),(11,23,43),(12,24,44),(13,17,45),(14,18,46),(15,19,47),(16,20,48)], [(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),(41,42,43,44,45,46,47,48)], [(2,8),(3,7),(4,6),(9,11),(12,16),(13,15),(17,19),(20,24),(21,23),(25,31),(26,30),(27,29),(34,40),(35,39),(36,38),(41,43),(44,48),(45,47)], [(1,32,37),(2,25,38),(3,26,39),(4,27,40),(5,28,33),(6,29,34),(7,30,35),(8,31,36),(9,21,41),(10,22,42),(11,23,43),(12,24,44),(13,17,45),(14,18,46),(15,19,47),(16,20,48)], [(1,18),(2,23),(3,20),(4,17),(5,22),(6,19),(7,24),(8,21),(9,31),(10,28),(11,25),(12,30),(13,27),(14,32),(15,29),(16,26),(33,42),(34,47),(35,44),(36,41),(37,46),(38,43),(39,48),(40,45)]])

54 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D3E4A4B4C6A6B6C6D6E6F6G6H6I6J6K6L···6Q6R6S8A8B12A12B12C12D12E12F12G12H12I24A···24H24I24J
order12222233333444666666666666···6668812121212121212121224···242424
size1144612112222612112224444668···81212412224446612124···41212

54 irreducible representations

dim11111111111111112222222222444444
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C3C6C6C6C6C6C6C6S3D4D4D6D6C3×S3C3×D4C3×D4S3×C6S3×C6C8⋊C22S3×D4D8⋊S3C3×C8⋊C22C3×S3×D4C3×D8⋊S3
kernelC3×D8⋊S3C3×C8⋊S3C3×C24⋊C2C3×D4⋊S3C3×D4.S3C32×D8C3×S3×D4C3×D42S3D8⋊S3C8⋊S3C24⋊C2D4⋊S3D4.S3C3×D8S3×D4D42S3C3×D8C3×Dic3S3×C6C24C3×D4D8Dic3D6C8D4C32C6C3C3C2C1
# reps11111111222222221111222224112224

Matrix representation of C3×D8⋊S3 in GL4(𝔽7) generated by

4000
0400
0040
0004
,
0623
5535
1340
6405
,
0526
2021
3301
1630
,
2511
3336
4346
2263
,
5122
0065
4356
5514
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[0,5,1,6,6,5,3,4,2,3,4,0,3,5,0,5],[0,2,3,1,5,0,3,6,2,2,0,3,6,1,1,0],[2,3,4,2,5,3,3,2,1,3,4,6,1,6,6,3],[5,0,4,5,1,0,3,5,2,6,5,1,2,5,6,4] >;

C3×D8⋊S3 in GAP, Magma, Sage, TeX

C_3\times D_8\rtimes S_3
% in TeX

G:=Group("C3xD8:S3");
// GroupNames label

G:=SmallGroup(288,682);
// by ID

G=gap.SmallGroup(288,682);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,1094,303,1271,648,102,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^8=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e=b^5,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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