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## G = C3×Q8⋊3S3order 144 = 24·32

### Direct product of C3 and Q8⋊3S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×Q8⋊3S3
 Chief series C1 — C3 — C6 — C3×C6 — S3×C6 — S3×C12 — C3×Q8⋊3S3
 Lower central C3 — C6 — C3×Q8⋊3S3
 Upper central C1 — C6 — C3×Q8

Generators and relations for C3×Q83S3
G = < a,b,c,d,e | a3=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >

Subgroups: 172 in 86 conjugacy classes, 46 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C4○D4, C3×S3, C3×C6, C4×S3, D12, C2×C12, C3×D4, C3×Q8, C3×Q8, C3×Dic3, C3×C12, S3×C6, Q83S3, C3×C4○D4, S3×C12, C3×D12, Q8×C32, C3×Q83S3
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C3×S3, C22×S3, C22×C6, S3×C6, Q83S3, C3×C4○D4, S3×C2×C6, C3×Q83S3

Smallest permutation representation of C3×Q83S3
On 48 points
Generators in S48
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 10 47)(6 11 48)(7 12 45)(8 9 46)(21 32 27)(22 29 28)(23 30 25)(24 31 26)(33 37 42)(34 38 43)(35 39 44)(36 40 41)
(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 37 7 39)(6 40 8 38)(9 43 11 41)(10 42 12 44)(13 27 15 25)(14 26 16 28)(17 29 19 31)(18 32 20 30)(33 45 35 47)(34 48 36 46)
(1 14 19)(2 15 20)(3 16 17)(4 13 18)(5 10 47)(6 11 48)(7 12 45)(8 9 46)(21 27 32)(22 28 29)(23 25 30)(24 26 31)(33 37 42)(34 38 43)(35 39 44)(36 40 41)
(1 34)(2 33)(3 36)(4 35)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 47)(22 46)(23 45)(24 48)

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

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

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

C3×Q83S3 is a maximal subgroup of
D126D6  D12.11D6  D12.12D6  D12.13D6  D12.25D6  D1215D6  D1216D6  C3×S3×C4○D4  Q8⋊C93S3  (Q8×He3)⋊C2  D363C6  He35D4⋊C2
C3×Q83S3 is a maximal quotient of
C3×Q8×Dic3  (Q8×He3)⋊C2  D363C6

45 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F ··· 6K 12A ··· 12F 12G 12H 12I 12J 12K ··· 12S order 1 2 2 2 2 3 3 3 3 3 4 4 4 4 4 6 6 6 6 6 6 ··· 6 12 ··· 12 12 12 12 12 12 ··· 12 size 1 1 6 6 6 1 1 2 2 2 2 2 2 3 3 1 1 2 2 2 6 ··· 6 2 ··· 2 3 3 3 3 4 ··· 4

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 D6 C4○D4 C3×S3 S3×C6 C3×C4○D4 Q8⋊3S3 C3×Q8⋊3S3 kernel C3×Q8⋊3S3 S3×C12 C3×D12 Q8×C32 Q8⋊3S3 C4×S3 D12 C3×Q8 C3×Q8 C12 C32 Q8 C4 C3 C3 C1 # reps 1 3 3 1 2 6 6 2 1 3 2 2 6 4 1 2

Matrix representation of C3×Q83S3 in GL4(𝔽7) generated by

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 3 3 3 3 1 4 4 2 2 4 1 3 3 3 6 6
,
 4 1 1 1 3 0 3 6 6 5 1 2 2 5 6 2
,
 4 0 0 0 1 1 6 6 0 2 4 2 6 1 1 3
,
 3 6 4 5 3 3 5 3 5 1 2 0 2 6 4 6
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[3,1,2,3,3,4,4,3,3,4,1,6,3,2,3,6],[4,3,6,2,1,0,5,5,1,3,1,6,1,6,2,2],[4,1,0,6,0,1,2,1,0,6,4,1,0,6,2,3],[3,3,5,2,6,3,1,6,4,5,2,4,5,3,0,6] >;

C3×Q83S3 in GAP, Magma, Sage, TeX

C_3\times Q_8\rtimes_3S_3
% in TeX

G:=Group("C3xQ8:3S3");
// GroupNames label

G:=SmallGroup(144,165);
// by ID

G=gap.SmallGroup(144,165);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-3,151,506,260,122,3461]);
// Polycyclic

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

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