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## G = C32⋊C32order 288 = 25·32

### The semidirect product of C32 and C32 acting via C32/C4=C8

Aliases: C32⋊C32, C4.2F9, (C3×C6).C16, C2.(C2.F9), (C3×C12).1C8, C324C8.C4, C322C16.1C2, SmallGroup(288,373)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C32⋊C32
 Chief series C1 — C32 — C3×C6 — C3×C12 — C32⋊4C8 — C32⋊2C16 — C32⋊C32
 Lower central C32 — C32⋊C32
 Upper central C1 — C4

Generators and relations for C32⋊C32
G = < a,b,c | a3=b3=c32=1, cac-1=ab=ba, cbc-1=a >

Smallest permutation representation of C32⋊C32
On 96 points
Generators in S96
(2 79 55)(3 80 56)(4 57 81)(6 59 83)(7 60 84)(8 85 61)(10 87 63)(11 88 64)(12 33 89)(14 35 91)(15 36 92)(16 93 37)(18 95 39)(19 96 40)(20 41 65)(22 43 67)(23 44 68)(24 69 45)(26 71 47)(27 72 48)(28 49 73)(30 51 75)(31 52 76)(32 77 53)
(1 78 54)(3 80 56)(4 81 57)(5 58 82)(7 60 84)(8 61 85)(9 86 62)(11 88 64)(12 89 33)(13 34 90)(15 36 92)(16 37 93)(17 94 38)(19 96 40)(20 65 41)(21 42 66)(23 44 68)(24 45 69)(25 70 46)(27 72 48)(28 73 49)(29 50 74)(31 52 76)(32 53 77)
(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 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)

G:=sub<Sym(96)| (2,79,55)(3,80,56)(4,57,81)(6,59,83)(7,60,84)(8,85,61)(10,87,63)(11,88,64)(12,33,89)(14,35,91)(15,36,92)(16,93,37)(18,95,39)(19,96,40)(20,41,65)(22,43,67)(23,44,68)(24,69,45)(26,71,47)(27,72,48)(28,49,73)(30,51,75)(31,52,76)(32,77,53), (1,78,54)(3,80,56)(4,81,57)(5,58,82)(7,60,84)(8,61,85)(9,86,62)(11,88,64)(12,89,33)(13,34,90)(15,36,92)(16,37,93)(17,94,38)(19,96,40)(20,65,41)(21,42,66)(23,44,68)(24,45,69)(25,70,46)(27,72,48)(28,73,49)(29,50,74)(31,52,76)(32,53,77), (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)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)>;

G:=Group( (2,79,55)(3,80,56)(4,57,81)(6,59,83)(7,60,84)(8,85,61)(10,87,63)(11,88,64)(12,33,89)(14,35,91)(15,36,92)(16,93,37)(18,95,39)(19,96,40)(20,41,65)(22,43,67)(23,44,68)(24,69,45)(26,71,47)(27,72,48)(28,49,73)(30,51,75)(31,52,76)(32,77,53), (1,78,54)(3,80,56)(4,81,57)(5,58,82)(7,60,84)(8,61,85)(9,86,62)(11,88,64)(12,89,33)(13,34,90)(15,36,92)(16,37,93)(17,94,38)(19,96,40)(20,65,41)(21,42,66)(23,44,68)(24,45,69)(25,70,46)(27,72,48)(28,73,49)(29,50,74)(31,52,76)(32,53,77), (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)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96) );

G=PermutationGroup([[(2,79,55),(3,80,56),(4,57,81),(6,59,83),(7,60,84),(8,85,61),(10,87,63),(11,88,64),(12,33,89),(14,35,91),(15,36,92),(16,93,37),(18,95,39),(19,96,40),(20,41,65),(22,43,67),(23,44,68),(24,69,45),(26,71,47),(27,72,48),(28,49,73),(30,51,75),(31,52,76),(32,77,53)], [(1,78,54),(3,80,56),(4,81,57),(5,58,82),(7,60,84),(8,61,85),(9,86,62),(11,88,64),(12,89,33),(13,34,90),(15,36,92),(16,37,93),(17,94,38),(19,96,40),(20,65,41),(21,42,66),(23,44,68),(24,45,69),(25,70,46),(27,72,48),(28,73,49),(29,50,74),(31,52,76),(32,53,77)], [(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),(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)]])

36 conjugacy classes

 class 1 2 3 4A 4B 6 8A 8B 8C 8D 12A 12B 16A ··· 16H 32A ··· 32P order 1 2 3 4 4 6 8 8 8 8 12 12 16 ··· 16 32 ··· 32 size 1 1 8 1 1 8 9 9 9 9 8 8 9 ··· 9 9 ··· 9

36 irreducible representations

 dim 1 1 1 1 1 1 8 8 8 type + + + - image C1 C2 C4 C8 C16 C32 F9 C2.F9 C32⋊C32 kernel C32⋊C32 C32⋊2C16 C32⋊4C8 C3×C12 C3×C6 C32 C4 C2 C1 # reps 1 1 2 4 8 16 1 1 2

Matrix representation of C32⋊C32 in GL9(𝔽97)

 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 96 0 0 0 0 0 0 0 1 96 0 0 0 0 0 0 0 0 24 0 1 0 0 0 64 42 73 0 96 96 0 0 0 62 19 2 0 0 0 96 96 0 0 0 0 95 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 0 96 1 0 0 0 0 0 0 0 96 0 0 0 0 0 0 0 0 0 0 0 96 0 0 0 0 0 0 0 1 96 0 0 0 0 0 72 17 73 0 96 96 0 0 0 8 72 0 24 1 0 0 0 0 17 79 0 95 0 0 1 0 0 17 79 0 95 0 0 0 1
,
 46 0 0 0 0 0 0 0 0 0 0 0 0 0 96 1 0 0 0 64 42 73 73 95 96 0 0 0 0 0 0 0 0 0 96 1 0 62 19 2 2 0 0 95 96 0 12 48 68 68 80 72 24 0 0 12 48 69 68 80 72 24 0 0 34 53 11 11 96 79 95 0 0 27 44 11 11 96 79 95 0

G:=sub<GL(9,GF(97))| [1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,64,62,0,0,0,1,0,0,0,42,19,0,0,0,0,0,1,0,73,2,0,0,0,0,96,96,24,0,0,95,0,0,0,0,0,0,96,0,0,0,0,0,0,0,1,96,0,0,0,0,0,0,0,0,0,96,1,0,0,0,0,0,0,0,96,0],[1,0,0,0,0,0,0,0,0,0,96,96,0,0,72,8,17,17,0,1,0,0,0,17,72,79,79,0,0,0,0,1,73,0,0,0,0,0,0,96,96,0,24,95,95,0,0,0,0,0,96,1,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[46,0,0,0,0,0,0,0,0,0,0,64,0,62,12,12,34,27,0,0,42,0,19,48,48,53,44,0,0,73,0,2,68,69,11,11,0,0,73,0,2,68,68,11,11,0,96,95,0,0,80,80,96,96,0,1,96,0,0,72,72,79,79,0,0,0,96,95,24,24,95,95,0,0,0,1,96,0,0,0,0] >;

C32⋊C32 in GAP, Magma, Sage, TeX

C_3^2\rtimes C_{32}
% in TeX

G:=Group("C3^2:C32");
// GroupNames label

G:=SmallGroup(288,373);
// by ID

G=gap.SmallGroup(288,373);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,14,36,58,80,4037,4716,691,10982,6285,2372]);
// Polycyclic

G:=Group<a,b,c|a^3=b^3=c^32=1,c*a*c^-1=a*b=b*a,c*b*c^-1=a>;
// generators/relations

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