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## G = C3×S3×Q8order 144 = 24·32

### Direct product of C3, S3 and Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×S3×Q8
 Chief series C1 — C3 — C6 — C3×C6 — S3×C6 — S3×C12 — C3×S3×Q8
 Lower central C3 — C6 — C3×S3×Q8
 Upper central C1 — C6 — C3×Q8

Generators and relations for C3×S3×Q8
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, C2×C4, Q8, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×Q8, C3×S3, C3×C6, Dic6, C4×S3, C2×C12, C3×Q8, C3×Q8, C3×Dic3, C3×C12, S3×C6, S3×Q8, C6×Q8, C3×Dic6, S3×C12, Q8×C32, C3×S3×Q8
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, D6, C2×C6, C2×Q8, C3×S3, C3×Q8, C22×S3, C22×C6, S3×C6, S3×Q8, C6×Q8, S3×C2×C6, C3×S3×Q8

Smallest permutation representation of C3×S3×Q8
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)]])

C3×S3×Q8 is a maximal subgroup of   D12.24D6  Dic6.22D6  D12.25D6  Dic6.26D6

45 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 6G 6H 6I 12A ··· 12F 12G ··· 12O 12P ··· 12U order 1 2 2 2 3 3 3 3 3 4 4 4 4 4 4 6 6 6 6 6 6 6 6 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 3 3 1 1 2 2 2 2 2 2 6 6 6 1 1 2 2 2 3 3 3 3 2 ··· 2 4 ··· 4 6 ··· 6

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + - + - image C1 C2 C2 C2 C3 C6 C6 C6 S3 Q8 D6 C3×S3 C3×Q8 S3×C6 S3×Q8 C3×S3×Q8 kernel C3×S3×Q8 C3×Dic6 S3×C12 Q8×C32 S3×Q8 Dic6 C4×S3 C3×Q8 C3×Q8 C3×S3 C12 Q8 S3 C4 C3 C1 # reps 1 3 3 1 2 6 6 2 1 2 3 2 4 6 1 2

Matrix representation of C3×S3×Q8 in GL4(𝔽7) generated by

 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 4 0 0 0 1 1 6 6 0 2 4 2 6 1 1 3
,
 5 4 2 0 4 2 6 1 1 0 4 5 1 2 5 3
,
 0 6 6 6 0 0 6 1 4 4 4 3 4 3 4 3
,
 2 3 3 3 2 6 0 4 5 1 3 5 3 5 2 3
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] >;

C3×S3×Q8 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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