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## G = C3×C32⋊4C8order 216 = 23·33

### Direct product of C3 and C32⋊4C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3×C32⋊4C8
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C32×C12 — C3×C32⋊4C8
 Lower central C32 — C3×C32⋊4C8
 Upper central C1 — C12

Generators and relations for C3×C324C8
G = < a,b,c,d | a3=b3=c3=d8=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 128 in 72 conjugacy classes, 38 normal (14 characteristic)
C1, C2, C3, C3, C3, C4, C6, C6, C6, C8, C32, C32, C32, C12, C12, C12, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, C33, C3×C12, C3×C12, C3×C12, C32×C6, C3×C3⋊C8, C324C8, C32×C12, C3×C324C8
Quotients: C1, C2, C3, C4, S3, C6, C8, Dic3, C12, C3×S3, C3⋊S3, C3⋊C8, C24, C3×Dic3, C3⋊Dic3, C3×C3⋊S3, C3×C3⋊C8, C324C8, C3×C3⋊Dic3, C3×C324C8

Smallest permutation representation of C3×C324C8
On 72 points
Generators in S72
(1 12 54)(2 13 55)(3 14 56)(4 15 49)(5 16 50)(6 9 51)(7 10 52)(8 11 53)(17 36 69)(18 37 70)(19 38 71)(20 39 72)(21 40 65)(22 33 66)(23 34 67)(24 35 68)(25 41 57)(26 42 58)(27 43 59)(28 44 60)(29 45 61)(30 46 62)(31 47 63)(32 48 64)
(1 68 47)(2 48 69)(3 70 41)(4 42 71)(5 72 43)(6 44 65)(7 66 45)(8 46 67)(9 60 21)(10 22 61)(11 62 23)(12 24 63)(13 64 17)(14 18 57)(15 58 19)(16 20 59)(25 56 37)(26 38 49)(27 50 39)(28 40 51)(29 52 33)(30 34 53)(31 54 35)(32 36 55)
(1 24 31)(2 32 17)(3 18 25)(4 26 19)(5 20 27)(6 28 21)(7 22 29)(8 30 23)(9 44 40)(10 33 45)(11 46 34)(12 35 47)(13 48 36)(14 37 41)(15 42 38)(16 39 43)(49 58 71)(50 72 59)(51 60 65)(52 66 61)(53 62 67)(54 68 63)(55 64 69)(56 70 57)
(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)| (1,12,54)(2,13,55)(3,14,56)(4,15,49)(5,16,50)(6,9,51)(7,10,52)(8,11,53)(17,36,69)(18,37,70)(19,38,71)(20,39,72)(21,40,65)(22,33,66)(23,34,67)(24,35,68)(25,41,57)(26,42,58)(27,43,59)(28,44,60)(29,45,61)(30,46,62)(31,47,63)(32,48,64), (1,68,47)(2,48,69)(3,70,41)(4,42,71)(5,72,43)(6,44,65)(7,66,45)(8,46,67)(9,60,21)(10,22,61)(11,62,23)(12,24,63)(13,64,17)(14,18,57)(15,58,19)(16,20,59)(25,56,37)(26,38,49)(27,50,39)(28,40,51)(29,52,33)(30,34,53)(31,54,35)(32,36,55), (1,24,31)(2,32,17)(3,18,25)(4,26,19)(5,20,27)(6,28,21)(7,22,29)(8,30,23)(9,44,40)(10,33,45)(11,46,34)(12,35,47)(13,48,36)(14,37,41)(15,42,38)(16,39,43)(49,58,71)(50,72,59)(51,60,65)(52,66,61)(53,62,67)(54,68,63)(55,64,69)(56,70,57), (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( (1,12,54)(2,13,55)(3,14,56)(4,15,49)(5,16,50)(6,9,51)(7,10,52)(8,11,53)(17,36,69)(18,37,70)(19,38,71)(20,39,72)(21,40,65)(22,33,66)(23,34,67)(24,35,68)(25,41,57)(26,42,58)(27,43,59)(28,44,60)(29,45,61)(30,46,62)(31,47,63)(32,48,64), (1,68,47)(2,48,69)(3,70,41)(4,42,71)(5,72,43)(6,44,65)(7,66,45)(8,46,67)(9,60,21)(10,22,61)(11,62,23)(12,24,63)(13,64,17)(14,18,57)(15,58,19)(16,20,59)(25,56,37)(26,38,49)(27,50,39)(28,40,51)(29,52,33)(30,34,53)(31,54,35)(32,36,55), (1,24,31)(2,32,17)(3,18,25)(4,26,19)(5,20,27)(6,28,21)(7,22,29)(8,30,23)(9,44,40)(10,33,45)(11,46,34)(12,35,47)(13,48,36)(14,37,41)(15,42,38)(16,39,43)(49,58,71)(50,72,59)(51,60,65)(52,66,61)(53,62,67)(54,68,63)(55,64,69)(56,70,57), (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([[(1,12,54),(2,13,55),(3,14,56),(4,15,49),(5,16,50),(6,9,51),(7,10,52),(8,11,53),(17,36,69),(18,37,70),(19,38,71),(20,39,72),(21,40,65),(22,33,66),(23,34,67),(24,35,68),(25,41,57),(26,42,58),(27,43,59),(28,44,60),(29,45,61),(30,46,62),(31,47,63),(32,48,64)], [(1,68,47),(2,48,69),(3,70,41),(4,42,71),(5,72,43),(6,44,65),(7,66,45),(8,46,67),(9,60,21),(10,22,61),(11,62,23),(12,24,63),(13,64,17),(14,18,57),(15,58,19),(16,20,59),(25,56,37),(26,38,49),(27,50,39),(28,40,51),(29,52,33),(30,34,53),(31,54,35),(32,36,55)], [(1,24,31),(2,32,17),(3,18,25),(4,26,19),(5,20,27),(6,28,21),(7,22,29),(8,30,23),(9,44,40),(10,33,45),(11,46,34),(12,35,47),(13,48,36),(14,37,41),(15,42,38),(16,39,43),(49,58,71),(50,72,59),(51,60,65),(52,66,61),(53,62,67),(54,68,63),(55,64,69),(56,70,57)], [(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 2 3A 3B 3C ··· 3N 4A 4B 6A 6B 6C ··· 6N 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12AB 24A ··· 24H order 1 2 3 3 3 ··· 3 4 4 6 6 6 ··· 6 8 8 8 8 12 12 12 12 12 ··· 12 24 ··· 24 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 9 9 9 9 1 1 1 1 2 ··· 2 9 ··· 9

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + - image C1 C2 C3 C4 C6 C8 C12 C24 S3 Dic3 C3×S3 C3⋊C8 C3×Dic3 C3×C3⋊C8 kernel C3×C32⋊4C8 C32×C12 C32⋊4C8 C32×C6 C3×C12 C33 C3×C6 C32 C3×C12 C3×C6 C12 C32 C6 C3 # reps 1 1 2 2 2 4 4 8 4 4 8 8 8 16

Matrix representation of C3×C324C8 in GL4(𝔽73) generated by

 64 0 0 0 0 64 0 0 0 0 8 0 0 0 0 8
,
 8 17 0 0 0 64 0 0 0 0 8 0 0 0 0 64
,
 1 0 0 0 0 1 0 0 0 0 64 0 0 0 0 8
,
 51 29 0 0 22 22 0 0 0 0 0 1 0 0 72 0
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,8,0,0,0,0,8],[8,0,0,0,17,64,0,0,0,0,8,0,0,0,0,64],[1,0,0,0,0,1,0,0,0,0,64,0,0,0,0,8],[51,22,0,0,29,22,0,0,0,0,0,72,0,0,1,0] >;

C3×C324C8 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes_4C_8
% in TeX

G:=Group("C3xC3^2:4C8");
// GroupNames label

G:=SmallGroup(216,83);
// by ID

G=gap.SmallGroup(216,83);
# by ID

G:=PCGroup([6,-2,-3,-2,-2,-3,-3,36,50,1444,5189]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^3=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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