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## G = C2×C33⋊9(C2×C4)  order 432 = 24·33

### Direct product of C2 and C33⋊9(C2×C4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C2×C33⋊9(C2×C4)
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C6×C3⋊S3 — C33⋊9(C2×C4) — C2×C33⋊9(C2×C4)
 Lower central C33 — C2×C33⋊9(C2×C4)
 Upper central C1 — C22

Generators and relations for C2×C339(C2×C4)
G = < a,b,c,d,e,f | a2=b3=c3=d3=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe=fbf-1=b-1, cd=dc, ce=ec, fcf-1=c-1, ede=d-1, df=fd, ef=fe >

Subgroups: 1208 in 290 conjugacy classes, 79 normal (15 characteristic)
C1, C2, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, C23, C32, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22×C4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C33, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C62, C62, C62, S3×C2×C4, C22×Dic3, C3×C3⋊S3, C32×C6, C32×C6, S3×Dic3, C6.D6, C6×Dic3, C2×C3⋊Dic3, S3×C2×C6, C22×C3⋊S3, C3×C3⋊Dic3, C6×C3⋊S3, C3×C62, C2×S3×Dic3, C2×C6.D6, C339(C2×C4), C6×C3⋊Dic3, C2×C6×C3⋊S3, C2×C339(C2×C4)
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C4×S3, C2×Dic3, C22×S3, S32, S3×C2×C4, C22×Dic3, S3×Dic3, C6.D6, C2×S32, C324D6, C2×S3×Dic3, C2×C6.D6, C339(C2×C4), C2×C324D6, C2×C339(C2×C4)

Smallest permutation representation of C2×C339(C2×C4)
On 48 points
Generators in S48
(1 11)(2 12)(3 9)(4 10)(5 24)(6 21)(7 22)(8 23)(13 31)(14 32)(15 29)(16 30)(17 25)(18 26)(19 27)(20 28)(33 43)(34 44)(35 41)(36 42)(37 45)(38 46)(39 47)(40 48)
(1 19 6)(2 7 20)(3 17 8)(4 5 18)(9 25 23)(10 24 26)(11 27 21)(12 22 28)(13 43 47)(14 48 44)(15 41 45)(16 46 42)(29 35 37)(30 38 36)(31 33 39)(32 40 34)
(1 19 6)(2 7 20)(3 17 8)(4 5 18)(9 25 23)(10 24 26)(11 27 21)(12 22 28)(13 47 43)(14 44 48)(15 45 41)(16 42 46)(29 37 35)(30 36 38)(31 39 33)(32 34 40)
(1 6 19)(2 7 20)(3 8 17)(4 5 18)(9 23 25)(10 24 26)(11 21 27)(12 22 28)(13 47 43)(14 48 44)(15 45 41)(16 46 42)(29 37 35)(30 38 36)(31 39 33)(32 40 34)
(1 35)(2 36)(3 33)(4 34)(5 40)(6 37)(7 38)(8 39)(9 43)(10 44)(11 41)(12 42)(13 25)(14 26)(15 27)(16 28)(17 31)(18 32)(19 29)(20 30)(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)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D ··· 3H 4A ··· 4H 6A ··· 6I 6J ··· 6X 6Y 6Z 6AA 6AB 12A ··· 12H order 1 2 2 2 2 2 2 2 3 3 3 3 ··· 3 4 ··· 4 6 ··· 6 6 ··· 6 6 6 6 6 12 ··· 12 size 1 1 1 1 9 9 9 9 2 2 2 4 ··· 4 9 ··· 9 2 ··· 2 4 ··· 4 18 18 18 18 18 ··· 18

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + - + + + - + + image C1 C2 C2 C2 C4 S3 S3 D6 Dic3 D6 D6 C4×S3 S32 S3×Dic3 C6.D6 C2×S32 C32⋊4D6 C33⋊9(C2×C4) C2×C32⋊4D6 kernel C2×C33⋊9(C2×C4) C33⋊9(C2×C4) C6×C3⋊Dic3 C2×C6×C3⋊S3 C6×C3⋊S3 C2×C3⋊Dic3 C22×C3⋊S3 C3⋊Dic3 C2×C3⋊S3 C2×C3⋊S3 C62 C3×C6 C2×C6 C6 C6 C6 C22 C2 C2 # reps 1 4 2 1 8 2 1 4 4 2 3 8 3 4 2 3 2 4 2

Matrix representation of C2×C339(C2×C4) in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 0 1 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 12 0 0 0 0 0 12
,
 8 0 0 0 0 0 5 5 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 5 8 0 0 0 0 0 8

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,12,12],[8,5,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,8,8] >;

C2×C339(C2×C4) in GAP, Magma, Sage, TeX

C_2\times C_3^3\rtimes_9(C_2\times C_4)
% in TeX

G:=Group("C2xC3^3:9(C2xC4)");
// GroupNames label

G:=SmallGroup(432,692);
// by ID

G=gap.SmallGroup(432,692);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,1124,571,2028,14118]);
// Polycyclic

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

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