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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 [×4], C3 [×2], C3, C4, C4 [×2], C22 [×6], S3 [×2], C6 [×2], C6 [×9], C8, C8, C2×C4 [×2], D4 [×2], D4 [×3], Q8, C23, C32, Dic3, Dic3, C12 [×2], C12 [×3], D6, D6 [×3], C2×C6 [×10], M4(2), D8, D8, SD16 [×2], C2×D4, C4○D4, C3×S3 [×2], C3×C6, C3×C6 [×2], C3⋊C8, C24 [×2], C24 [×2], Dic6, C4×S3, D12, C2×Dic3, C3⋊D4 [×2], C2×C12 [×2], C3×D4 [×4], C3×D4 [×5], C3×Q8, C22×S3, C22×C6, C8⋊C22, C3×Dic3, C3×Dic3, C3×C12, S3×C6, S3×C6 [×3], C62 [×2], C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×M4(2), C3×D8 [×2], C3×D8 [×2], C3×SD16 [×2], S3×D4, D42S3, C6×D4, C3×C4○D4, C3×C3⋊C8, C3×C24, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4 [×2], D4×C32 [×2], 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 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C3×S3, C3×D4 [×2], C22×S3, C22×C6, C8⋊C22, S3×C6 [×3], 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 40 32)(2 33 25)(3 34 26)(4 35 27)(5 36 28)(6 37 29)(7 38 30)(8 39 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)(33 39)(34 38)(35 37)(41 43)(44 48)(45 47)
(1 32 40)(2 25 33)(3 26 34)(4 27 35)(5 28 36)(6 29 37)(7 30 38)(8 31 39)(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 43)(34 48)(35 45)(36 42)(37 47)(38 44)(39 41)(40 46)

G:=sub<Sym(48)| (1,40,32)(2,33,25)(3,34,26)(4,35,27)(5,36,28)(6,37,29)(7,38,30)(8,39,31)(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)(33,39)(34,38)(35,37)(41,43)(44,48)(45,47), (1,32,40)(2,25,33)(3,26,34)(4,27,35)(5,28,36)(6,29,37)(7,30,38)(8,31,39)(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)(33,43)(34,48)(35,45)(36,42)(37,47)(38,44)(39,41)(40,46)>;

G:=Group( (1,40,32)(2,33,25)(3,34,26)(4,35,27)(5,36,28)(6,37,29)(7,38,30)(8,39,31)(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)(33,39)(34,38)(35,37)(41,43)(44,48)(45,47), (1,32,40)(2,25,33)(3,26,34)(4,27,35)(5,28,36)(6,29,37)(7,30,38)(8,31,39)(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)(33,43)(34,48)(35,45)(36,42)(37,47)(38,44)(39,41)(40,46) );

G=PermutationGroup([(1,40,32),(2,33,25),(3,34,26),(4,35,27),(5,36,28),(6,37,29),(7,38,30),(8,39,31),(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),(33,39),(34,38),(35,37),(41,43),(44,48),(45,47)], [(1,32,40),(2,25,33),(3,26,34),(4,27,35),(5,28,36),(6,29,37),(7,30,38),(8,31,39),(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),(33,43),(34,48),(35,45),(36,42),(37,47),(38,44),(39,41),(40,46)])

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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