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G = C3xS3xQ8order 144 = 24·32

Direct product of C3, S3 and Q8

direct product, metabelian, supersoluble, monomial

Aliases: C3xS3xQ8, Dic6:4C6, C12.38D6, C3:2(C6xQ8), C4.6(S3xC6), (C3xQ8):4C6, (C4xS3).1C6, C32:7(C2xQ8), C12.6(C2xC6), D6.5(C2xC6), (S3xC12).3C2, (C3xDic6):9C2, C6.7(C22xC6), (Q8xC32):3C2, (C3xC6).25C23, C6.46(C22xS3), Dic3.4(C2xC6), (S3xC6).14C22, (C3xC12).22C22, (C3xDic3).12C22, C2.8(S3xC2xC6), SmallGroup(144,164)

Series: Derived Chief Lower central Upper central

C1C6 — C3xS3xQ8
C1C3C6C3xC6S3xC6S3xC12 — C3xS3xQ8
C3C6 — C3xS3xQ8
C1C6C3xQ8

Generators and relations for C3xS3xQ8
 G = < a,b,c,d,e | a3=b3=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 140 in 82 conjugacy classes, 50 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2xC4, Q8, Q8, C32, Dic3, C12, C12, D6, C2xC6, C2xQ8, C3xS3, C3xC6, Dic6, C4xS3, C2xC12, C3xQ8, C3xQ8, C3xDic3, C3xC12, S3xC6, S3xQ8, C6xQ8, C3xDic6, S3xC12, Q8xC32, C3xS3xQ8
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, D6, C2xC6, C2xQ8, C3xS3, C3xQ8, C22xS3, C22xC6, S3xC6, S3xQ8, C6xQ8, S3xC2xC6, C3xS3xQ8

Smallest permutation representation of C3xS3xQ8
On 48 points
Generators in S48
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 10 45)(6 11 46)(7 12 47)(8 9 48)(21 25 30)(22 26 31)(23 27 32)(24 28 29)(33 42 37)(34 43 38)(35 44 39)(36 41 40)
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 45 10)(6 46 11)(7 47 12)(8 48 9)(21 25 30)(22 26 31)(23 27 32)(24 28 29)(33 37 42)(34 38 43)(35 39 44)(36 40 41)
(1 34)(2 35)(3 36)(4 33)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 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)
(1 24 3 22)(2 23 4 21)(5 44 7 42)(6 43 8 41)(9 40 11 38)(10 39 12 37)(13 30 15 32)(14 29 16 31)(17 26 19 28)(18 25 20 27)(33 45 35 47)(34 48 36 46)

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

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

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

C3xS3xQ8 is a maximal subgroup of   D12.24D6  Dic6.22D6  D12.25D6  Dic6.26D6

45 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D4E4F6A6B6C6D6E6F6G6H6I12A···12F12G···12O12P···12U
order12223333344444466666666612···1212···1212···12
size1133112222226661122233332···24···46···6

45 irreducible representations

dim1111111122222244
type+++++-+-
imageC1C2C2C2C3C6C6C6S3Q8D6C3xS3C3xQ8S3xC6S3xQ8C3xS3xQ8
kernelC3xS3xQ8C3xDic6S3xC12Q8xC32S3xQ8Dic6C4xS3C3xQ8C3xQ8C3xS3C12Q8S3C4C3C1
# reps1331266212324612

Matrix representation of C3xS3xQ8 in GL4(F7) generated by

2000
0200
0020
0002
,
4000
1166
0242
6113
,
5420
4261
1045
1253
,
0666
0061
4443
4343
,
2333
2604
5135
3523
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[4,1,0,6,0,1,2,1,0,6,4,1,0,6,2,3],[5,4,1,1,4,2,0,2,2,6,4,5,0,1,5,3],[0,0,4,4,6,0,4,3,6,6,4,4,6,1,3,3],[2,2,5,3,3,6,1,5,3,0,3,2,3,4,5,3] >;

C3xS3xQ8 in GAP, Magma, Sage, TeX

C_3\times S_3\times Q_8
% in TeX

G:=Group("C3xS3xQ8");
// GroupNames label

G:=SmallGroup(144,164);
// by ID

G=gap.SmallGroup(144,164);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-3,151,260,122,3461]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^4=1,e^2=d^2,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^-1=d^-1>;
// generators/relations

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