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## G = C32×C3⋊C8order 216 = 23·33

### Direct product of C32 and C3⋊C8

Series: Derived Chief Lower central Upper central

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

Generators and relations for C32×C3⋊C8
G = < a,b,c,d | a3=b3=c3=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 108 in 72 conjugacy classes, 42 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, C3×C24, C32×C12, C32×C3⋊C8
Quotients: C1, C2, C3, C4, S3, C6, C8, C32, Dic3, C12, C3×S3, C3×C6, C3⋊C8, C24, C3×Dic3, C3×C12, S3×C32, C3×C3⋊C8, C3×C24, C32×Dic3, C32×C3⋊C8

Smallest permutation representation of C32×C3⋊C8
On 72 points
Generators in S72
(1 45 13)(2 46 14)(3 47 15)(4 48 16)(5 41 9)(6 42 10)(7 43 11)(8 44 12)(17 72 58)(18 65 59)(19 66 60)(20 67 61)(21 68 62)(22 69 63)(23 70 64)(24 71 57)(25 53 39)(26 54 40)(27 55 33)(28 56 34)(29 49 35)(30 50 36)(31 51 37)(32 52 38)
(1 51 71)(2 52 72)(3 53 65)(4 54 66)(5 55 67)(6 56 68)(7 49 69)(8 50 70)(9 27 20)(10 28 21)(11 29 22)(12 30 23)(13 31 24)(14 32 17)(15 25 18)(16 26 19)(33 61 41)(34 62 42)(35 63 43)(36 64 44)(37 57 45)(38 58 46)(39 59 47)(40 60 48)
(1 51 71)(2 72 52)(3 53 65)(4 66 54)(5 55 67)(6 68 56)(7 49 69)(8 70 50)(9 27 20)(10 21 28)(11 29 22)(12 23 30)(13 31 24)(14 17 32)(15 25 18)(16 19 26)(33 61 41)(34 42 62)(35 63 43)(36 44 64)(37 57 45)(38 46 58)(39 59 47)(40 48 60)
(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)| (1,45,13)(2,46,14)(3,47,15)(4,48,16)(5,41,9)(6,42,10)(7,43,11)(8,44,12)(17,72,58)(18,65,59)(19,66,60)(20,67,61)(21,68,62)(22,69,63)(23,70,64)(24,71,57)(25,53,39)(26,54,40)(27,55,33)(28,56,34)(29,49,35)(30,50,36)(31,51,37)(32,52,38), (1,51,71)(2,52,72)(3,53,65)(4,54,66)(5,55,67)(6,56,68)(7,49,69)(8,50,70)(9,27,20)(10,28,21)(11,29,22)(12,30,23)(13,31,24)(14,32,17)(15,25,18)(16,26,19)(33,61,41)(34,62,42)(35,63,43)(36,64,44)(37,57,45)(38,58,46)(39,59,47)(40,60,48), (1,51,71)(2,72,52)(3,53,65)(4,66,54)(5,55,67)(6,68,56)(7,49,69)(8,70,50)(9,27,20)(10,21,28)(11,29,22)(12,23,30)(13,31,24)(14,17,32)(15,25,18)(16,19,26)(33,61,41)(34,42,62)(35,63,43)(36,44,64)(37,57,45)(38,46,58)(39,59,47)(40,48,60), (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( (1,45,13)(2,46,14)(3,47,15)(4,48,16)(5,41,9)(6,42,10)(7,43,11)(8,44,12)(17,72,58)(18,65,59)(19,66,60)(20,67,61)(21,68,62)(22,69,63)(23,70,64)(24,71,57)(25,53,39)(26,54,40)(27,55,33)(28,56,34)(29,49,35)(30,50,36)(31,51,37)(32,52,38), (1,51,71)(2,52,72)(3,53,65)(4,54,66)(5,55,67)(6,56,68)(7,49,69)(8,50,70)(9,27,20)(10,28,21)(11,29,22)(12,30,23)(13,31,24)(14,32,17)(15,25,18)(16,26,19)(33,61,41)(34,62,42)(35,63,43)(36,64,44)(37,57,45)(38,58,46)(39,59,47)(40,60,48), (1,51,71)(2,72,52)(3,53,65)(4,66,54)(5,55,67)(6,68,56)(7,49,69)(8,70,50)(9,27,20)(10,21,28)(11,29,22)(12,23,30)(13,31,24)(14,17,32)(15,25,18)(16,19,26)(33,61,41)(34,42,62)(35,63,43)(36,44,64)(37,57,45)(38,46,58)(39,59,47)(40,48,60), (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([[(1,45,13),(2,46,14),(3,47,15),(4,48,16),(5,41,9),(6,42,10),(7,43,11),(8,44,12),(17,72,58),(18,65,59),(19,66,60),(20,67,61),(21,68,62),(22,69,63),(23,70,64),(24,71,57),(25,53,39),(26,54,40),(27,55,33),(28,56,34),(29,49,35),(30,50,36),(31,51,37),(32,52,38)], [(1,51,71),(2,52,72),(3,53,65),(4,54,66),(5,55,67),(6,56,68),(7,49,69),(8,50,70),(9,27,20),(10,28,21),(11,29,22),(12,30,23),(13,31,24),(14,32,17),(15,25,18),(16,26,19),(33,61,41),(34,62,42),(35,63,43),(36,64,44),(37,57,45),(38,58,46),(39,59,47),(40,60,48)], [(1,51,71),(2,72,52),(3,53,65),(4,66,54),(5,55,67),(6,68,56),(7,49,69),(8,70,50),(9,27,20),(10,21,28),(11,29,22),(12,23,30),(13,31,24),(14,17,32),(15,25,18),(16,19,26),(33,61,41),(34,42,62),(35,63,43),(36,44,64),(37,57,45),(38,46,58),(39,59,47),(40,48,60)], [(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)]])

C32×C3⋊C8 is a maximal subgroup of
C12.69S32  C338M4(2)  C339M4(2)  C338D8  C3316SD16  C3317SD16  C338Q16  S3×C3×C24

108 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Q 4A 4B 6A ··· 6H 6I ··· 6Q 8A 8B 8C 8D 12A ··· 12P 12Q ··· 12AH 24A ··· 24AF order 1 2 3 ··· 3 3 ··· 3 4 4 6 ··· 6 6 ··· 6 8 8 8 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 1 ··· 1 2 ··· 2 1 1 1 ··· 1 2 ··· 2 3 3 3 3 1 ··· 1 2 ··· 2 3 ··· 3

108 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 C32×C3⋊C8 C32×C12 C3×C3⋊C8 C32×C6 C3×C12 C33 C3×C6 C32 C3×C12 C3×C6 C12 C32 C6 C3 # reps 1 1 8 2 8 4 16 32 1 1 8 2 8 16

Matrix representation of C32×C3⋊C8 in GL3(𝔽73) generated by

 8 0 0 0 1 0 0 0 1
,
 64 0 0 0 8 0 0 0 8
,
 1 0 0 0 8 0 0 0 64
,
 22 0 0 0 0 1 0 27 0
G:=sub<GL(3,GF(73))| [8,0,0,0,1,0,0,0,1],[64,0,0,0,8,0,0,0,8],[1,0,0,0,8,0,0,0,64],[22,0,0,0,0,27,0,1,0] >;

C32×C3⋊C8 in GAP, Magma, Sage, TeX

C_3^2\times C_3\rtimes C_8
% in TeX

G:=Group("C3^2xC3:C8");
// GroupNames label

G:=SmallGroup(216,82);
// by ID

G=gap.SmallGroup(216,82);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,-2,-3,108,69,5189]);
// 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,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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