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## G = C2×C33⋊8(C2×C4)  order 432 = 24·33

### Direct product of C2 and C33⋊8(C2×C4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C2×C33⋊8(C2×C4)
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C32×Dic3 — C33⋊8(C2×C4) — C2×C33⋊8(C2×C4)
 Lower central C33 — C2×C33⋊8(C2×C4)
 Upper central C1 — C22

Generators and relations for C2×C338(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, ece=fcf-1=c-1, ede=d-1, df=fd, ef=fe >

Subgroups: 2808 in 452 conjugacy classes, 100 normal (18 characteristic)
C1, C2, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, C23, C32, C32, C32, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22×C4, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C22×S3, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, C62, C62, S3×C2×C4, C33⋊C2, C32×C6, C32×C6, C6.D6, C6×Dic3, C6×Dic3, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C32×Dic3, C3×C3⋊Dic3, C2×C33⋊C2, C3×C62, C2×C6.D6, C2×C4×C3⋊S3, C338(C2×C4), Dic3×C3×C6, C6×C3⋊Dic3, C22×C33⋊C2, C2×C338(C2×C4)
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C3⋊S3, C4×S3, C22×S3, S32, C2×C3⋊S3, S3×C2×C4, C6.D6, C4×C3⋊S3, C2×S32, C22×C3⋊S3, S3×C3⋊S3, C2×C6.D6, C2×C4×C3⋊S3, C338(C2×C4), C2×S3×C3⋊S3, C2×C338(C2×C4)

Smallest permutation representation of C2×C338(C2×C4)
On 72 points
Generators in S72
(1 40)(2 37)(3 38)(4 39)(5 46)(6 47)(7 48)(8 45)(9 18)(10 19)(11 20)(12 17)(13 35)(14 36)(15 33)(16 34)(21 25)(22 26)(23 27)(24 28)(29 58)(30 59)(31 60)(32 57)(41 62)(42 63)(43 64)(44 61)(49 71)(50 72)(51 69)(52 70)(53 66)(54 67)(55 68)(56 65)
(1 18 8)(2 5 19)(3 20 6)(4 7 17)(9 45 40)(10 37 46)(11 47 38)(12 39 48)(13 49 64)(14 61 50)(15 51 62)(16 63 52)(21 55 57)(22 58 56)(23 53 59)(24 60 54)(25 68 32)(26 29 65)(27 66 30)(28 31 67)(33 69 41)(34 42 70)(35 71 43)(36 44 72)
(1 15 26)(2 27 16)(3 13 28)(4 25 14)(5 66 63)(6 64 67)(7 68 61)(8 62 65)(9 69 58)(10 59 70)(11 71 60)(12 57 72)(17 32 50)(18 51 29)(19 30 52)(20 49 31)(21 36 39)(22 40 33)(23 34 37)(24 38 35)(41 56 45)(42 46 53)(43 54 47)(44 48 55)
(1 15 26)(2 16 27)(3 13 28)(4 14 25)(5 63 66)(6 64 67)(7 61 68)(8 62 65)(9 69 58)(10 70 59)(11 71 60)(12 72 57)(17 50 32)(18 51 29)(19 52 30)(20 49 31)(21 39 36)(22 40 33)(23 37 34)(24 38 35)(41 56 45)(42 53 46)(43 54 47)(44 55 48)
(1 38)(2 39)(3 40)(4 37)(5 12)(6 9)(7 10)(8 11)(13 22)(14 23)(15 24)(16 21)(17 46)(18 47)(19 48)(20 45)(25 34)(26 35)(27 36)(28 33)(29 43)(30 44)(31 41)(32 42)(49 56)(50 53)(51 54)(52 55)(57 63)(58 64)(59 61)(60 62)(65 71)(66 72)(67 69)(68 70)
(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)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)(65 66 67 68)(69 70 71 72)

G:=sub<Sym(72)| (1,40)(2,37)(3,38)(4,39)(5,46)(6,47)(7,48)(8,45)(9,18)(10,19)(11,20)(12,17)(13,35)(14,36)(15,33)(16,34)(21,25)(22,26)(23,27)(24,28)(29,58)(30,59)(31,60)(32,57)(41,62)(42,63)(43,64)(44,61)(49,71)(50,72)(51,69)(52,70)(53,66)(54,67)(55,68)(56,65), (1,18,8)(2,5,19)(3,20,6)(4,7,17)(9,45,40)(10,37,46)(11,47,38)(12,39,48)(13,49,64)(14,61,50)(15,51,62)(16,63,52)(21,55,57)(22,58,56)(23,53,59)(24,60,54)(25,68,32)(26,29,65)(27,66,30)(28,31,67)(33,69,41)(34,42,70)(35,71,43)(36,44,72), (1,15,26)(2,27,16)(3,13,28)(4,25,14)(5,66,63)(6,64,67)(7,68,61)(8,62,65)(9,69,58)(10,59,70)(11,71,60)(12,57,72)(17,32,50)(18,51,29)(19,30,52)(20,49,31)(21,36,39)(22,40,33)(23,34,37)(24,38,35)(41,56,45)(42,46,53)(43,54,47)(44,48,55), (1,15,26)(2,16,27)(3,13,28)(4,14,25)(5,63,66)(6,64,67)(7,61,68)(8,62,65)(9,69,58)(10,70,59)(11,71,60)(12,72,57)(17,50,32)(18,51,29)(19,52,30)(20,49,31)(21,39,36)(22,40,33)(23,37,34)(24,38,35)(41,56,45)(42,53,46)(43,54,47)(44,55,48), (1,38)(2,39)(3,40)(4,37)(5,12)(6,9)(7,10)(8,11)(13,22)(14,23)(15,24)(16,21)(17,46)(18,47)(19,48)(20,45)(25,34)(26,35)(27,36)(28,33)(29,43)(30,44)(31,41)(32,42)(49,56)(50,53)(51,54)(52,55)(57,63)(58,64)(59,61)(60,62)(65,71)(66,72)(67,69)(68,70), (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)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64)(65,66,67,68)(69,70,71,72)>;

G:=Group( (1,40)(2,37)(3,38)(4,39)(5,46)(6,47)(7,48)(8,45)(9,18)(10,19)(11,20)(12,17)(13,35)(14,36)(15,33)(16,34)(21,25)(22,26)(23,27)(24,28)(29,58)(30,59)(31,60)(32,57)(41,62)(42,63)(43,64)(44,61)(49,71)(50,72)(51,69)(52,70)(53,66)(54,67)(55,68)(56,65), (1,18,8)(2,5,19)(3,20,6)(4,7,17)(9,45,40)(10,37,46)(11,47,38)(12,39,48)(13,49,64)(14,61,50)(15,51,62)(16,63,52)(21,55,57)(22,58,56)(23,53,59)(24,60,54)(25,68,32)(26,29,65)(27,66,30)(28,31,67)(33,69,41)(34,42,70)(35,71,43)(36,44,72), (1,15,26)(2,27,16)(3,13,28)(4,25,14)(5,66,63)(6,64,67)(7,68,61)(8,62,65)(9,69,58)(10,59,70)(11,71,60)(12,57,72)(17,32,50)(18,51,29)(19,30,52)(20,49,31)(21,36,39)(22,40,33)(23,34,37)(24,38,35)(41,56,45)(42,46,53)(43,54,47)(44,48,55), (1,15,26)(2,16,27)(3,13,28)(4,14,25)(5,63,66)(6,64,67)(7,61,68)(8,62,65)(9,69,58)(10,70,59)(11,71,60)(12,72,57)(17,50,32)(18,51,29)(19,52,30)(20,49,31)(21,39,36)(22,40,33)(23,37,34)(24,38,35)(41,56,45)(42,53,46)(43,54,47)(44,55,48), (1,38)(2,39)(3,40)(4,37)(5,12)(6,9)(7,10)(8,11)(13,22)(14,23)(15,24)(16,21)(17,46)(18,47)(19,48)(20,45)(25,34)(26,35)(27,36)(28,33)(29,43)(30,44)(31,41)(32,42)(49,56)(50,53)(51,54)(52,55)(57,63)(58,64)(59,61)(60,62)(65,71)(66,72)(67,69)(68,70), (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)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64)(65,66,67,68)(69,70,71,72) );

G=PermutationGroup([[(1,40),(2,37),(3,38),(4,39),(5,46),(6,47),(7,48),(8,45),(9,18),(10,19),(11,20),(12,17),(13,35),(14,36),(15,33),(16,34),(21,25),(22,26),(23,27),(24,28),(29,58),(30,59),(31,60),(32,57),(41,62),(42,63),(43,64),(44,61),(49,71),(50,72),(51,69),(52,70),(53,66),(54,67),(55,68),(56,65)], [(1,18,8),(2,5,19),(3,20,6),(4,7,17),(9,45,40),(10,37,46),(11,47,38),(12,39,48),(13,49,64),(14,61,50),(15,51,62),(16,63,52),(21,55,57),(22,58,56),(23,53,59),(24,60,54),(25,68,32),(26,29,65),(27,66,30),(28,31,67),(33,69,41),(34,42,70),(35,71,43),(36,44,72)], [(1,15,26),(2,27,16),(3,13,28),(4,25,14),(5,66,63),(6,64,67),(7,68,61),(8,62,65),(9,69,58),(10,59,70),(11,71,60),(12,57,72),(17,32,50),(18,51,29),(19,30,52),(20,49,31),(21,36,39),(22,40,33),(23,34,37),(24,38,35),(41,56,45),(42,46,53),(43,54,47),(44,48,55)], [(1,15,26),(2,16,27),(3,13,28),(4,14,25),(5,63,66),(6,64,67),(7,61,68),(8,62,65),(9,69,58),(10,70,59),(11,71,60),(12,72,57),(17,50,32),(18,51,29),(19,52,30),(20,49,31),(21,39,36),(22,40,33),(23,37,34),(24,38,35),(41,56,45),(42,53,46),(43,54,47),(44,55,48)], [(1,38),(2,39),(3,40),(4,37),(5,12),(6,9),(7,10),(8,11),(13,22),(14,23),(15,24),(16,21),(17,46),(18,47),(19,48),(20,45),(25,34),(26,35),(27,36),(28,33),(29,43),(30,44),(31,41),(32,42),(49,56),(50,53),(51,54),(52,55),(57,63),(58,64),(59,61),(60,62),(65,71),(66,72),(67,69),(68,70)], [(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),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64),(65,66,67,68),(69,70,71,72)]])

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A ··· 3E 3F 3G 3H 3I 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6O 6P ··· 6AA 12A ··· 12P 12Q 12R 12S 12T order 1 2 2 2 2 2 2 2 3 ··· 3 3 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 12 12 12 12 size 1 1 1 1 27 27 27 27 2 ··· 2 4 4 4 4 3 3 3 3 9 9 9 9 2 ··· 2 4 ··· 4 6 ··· 6 18 18 18 18

72 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 S3 S3 D6 D6 D6 C4×S3 S32 C6.D6 C2×S32 kernel C2×C33⋊8(C2×C4) C33⋊8(C2×C4) Dic3×C3×C6 C6×C3⋊Dic3 C22×C33⋊C2 C2×C33⋊C2 C6×Dic3 C2×C3⋊Dic3 C3×Dic3 C3⋊Dic3 C62 C3×C6 C2×C6 C6 C6 # reps 1 4 1 1 1 8 4 1 8 2 5 20 4 8 4

Matrix representation of C2×C338(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
,
 12 1 0 0 0 0 12 0 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
,
 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
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 12 12

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],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[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],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,12,0,0,0,0,0,12] >;

C2×C338(C2×C4) in GAP, Magma, Sage, TeX

C_2\times C_3^3\rtimes_8(C_2\times C_4)
% in TeX

G:=Group("C2xC3^3:8(C2xC4)");
// GroupNames label

G:=SmallGroup(432,679);
// by ID

G=gap.SmallGroup(432,679);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,64,571,2028,14118]);
// 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,e*c*e=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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