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G = C3xS3xD8order 288 = 25·32

Direct product of C3, S3 and D8

direct product, metabelian, supersoluble, monomial

Aliases: C3xS3xD8, D24:4C6, C24:18D6, C8:4(S3xC6), C3:2(C6xD8), (S3xC8):1C6, C24:2(C2xC6), D4:S3:1C6, (C3xD8):2C6, (S3xD4):1C6, D4:1(S3xC6), (S3xC24):5C2, (C3xD4):12D6, D12:1(C2xC6), C6.27(C6xD4), C32:11(C2xD8), (C3xD24):12C2, (C3xC24):8C22, D6.12(C3xD4), (S3xC6).48D4, C6.187(S3xD4), (C32xD8):3C2, C12.1(C22xC6), Dic3.3(C3xD4), (C3xD12):10C22, (C3xC12).72C23, (C3xDic3).30D4, (D4xC32):5C22, (S3xC12).47C22, C12.152(C22xS3), C3:C8:5(C2xC6), (C3xS3xD4):4C2, C4.1(S3xC2xC6), C2.15(C3xS3xD4), (C3xD4:S3):9C2, (C3xD4):1(C2xC6), (C3xC3:C8):31C22, (C4xS3).7(C2xC6), (C3xC6).215(C2xD4), SmallGroup(288,681)

Series: Derived Chief Lower central Upper central

C1C12 — C3xS3xD8
C1C3C6C12C3xC12S3xC12C3xS3xD4 — C3xS3xD8
C3C6C12 — C3xS3xD8
C1C6C12C3xD8

Generators and relations for C3xS3xD8
 G = < a,b,c,d,e | a3=b3=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 522 in 163 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, S3, C6, C6, C8, C8, C2xC4, D4, D4, C23, C32, Dic3, C12, C12, D6, D6, C2xC6, C2xC8, D8, D8, C2xD4, C3xS3, C3xS3, C3xC6, C3xC6, C3:C8, C24, C24, C4xS3, D12, C3:D4, C2xC12, C3xD4, C3xD4, C22xS3, C22xC6, C2xD8, C3xDic3, C3xC12, S3xC6, S3xC6, C62, S3xC8, D24, D4:S3, C2xC24, C3xD8, C3xD8, S3xD4, C6xD4, C3xC3:C8, C3xC24, S3xC12, C3xD12, C3xC3:D4, D4xC32, S3xC2xC6, S3xD8, C6xD8, S3xC24, C3xD24, C3xD4:S3, C32xD8, C3xS3xD4, C3xS3xD8
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2xC6, D8, C2xD4, C3xS3, C3xD4, C22xS3, C22xC6, C2xD8, S3xC6, C3xD8, S3xD4, C6xD4, S3xC2xC6, S3xD8, C6xD8, C3xS3xD4, C3xS3xD8

Smallest permutation representation of C3xS3xD8
On 48 points
Generators in S48
(1 36 27)(2 37 28)(3 38 29)(4 39 30)(5 40 31)(6 33 32)(7 34 25)(8 35 26)(9 21 45)(10 22 46)(11 23 47)(12 24 48)(13 17 41)(14 18 42)(15 19 43)(16 20 44)
(1 36 27)(2 37 28)(3 38 29)(4 39 30)(5 40 31)(6 33 32)(7 34 25)(8 35 26)(9 45 21)(10 46 22)(11 47 23)(12 48 24)(13 41 17)(14 42 18)(15 43 19)(16 44 20)
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)
(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 29)(26 28)(30 32)(33 39)(34 38)(35 37)(41 43)(44 48)(45 47)

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

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

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

63 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E4A4B6A6B6C6D6E6F6G6H6I6J6K6L6M6N···6S6T6U6V6W8A8B8C8D12A12B12C12D12E12F12G24A24B24C24D24E···24J24K24L24M24N
order12222222333334466666666666666···666668888121212121212122424242424···2424242424
size1133441212112222611222333344448···8121212122266224446622224···46666

63 irreducible representations

dim1111111111112222222222224444
type++++++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6S3D4D4D6D6D8C3xS3C3xD4C3xD4S3xC6S3xC6C3xD8S3xD4S3xD8C3xS3xD4C3xS3xD8
kernelC3xS3xD8S3xC24C3xD24C3xD4:S3C32xD8C3xS3xD4S3xD8S3xC8D24D4:S3C3xD8S3xD4C3xD8C3xDic3S3xC6C24C3xD4C3xS3D8Dic3D6C8D4S3C6C3C2C1
# reps1112122224241111242222481224

Matrix representation of C3xS3xD8 in GL4(F7) generated by

4000
0400
0040
0004
,
2331
6114
2206
4322
,
4253
6214
0654
5033
,
3302
1311
2263
3423
,
2061
6466
2412
6430
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,6,2,4,3,1,2,3,3,1,0,2,1,4,6,2],[4,6,0,5,2,2,6,0,5,1,5,3,3,4,4,3],[3,1,2,3,3,3,2,4,0,1,6,2,2,1,3,3],[2,6,2,6,0,4,4,4,6,6,1,3,1,6,2,0] >;

C3xS3xD8 in GAP, Magma, Sage, TeX

C_3\times S_3\times D_8
% in TeX

G:=Group("C3xS3xD8");
// GroupNames label

G:=SmallGroup(288,681);
// by ID

G=gap.SmallGroup(288,681);
# by ID

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

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

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