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## G = C3×C3⋊C32order 288 = 25·32

### Direct product of C3 and C3⋊C32

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C3×C3⋊C32
 Chief series C1 — C3 — C6 — C12 — C24 — C48 — C3×C48 — C3×C3⋊C32
 Lower central C3 — C3×C3⋊C32
 Upper central C1 — C48

Generators and relations for C3×C3⋊C32
G = < a,b,c | a3=b3=c32=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C3×C3⋊C32
On 96 points
Generators in S96
(1 42 77)(2 43 78)(3 44 79)(4 45 80)(5 46 81)(6 47 82)(7 48 83)(8 49 84)(9 50 85)(10 51 86)(11 52 87)(12 53 88)(13 54 89)(14 55 90)(15 56 91)(16 57 92)(17 58 93)(18 59 94)(19 60 95)(20 61 96)(21 62 65)(22 63 66)(23 64 67)(24 33 68)(25 34 69)(26 35 70)(27 36 71)(28 37 72)(29 38 73)(30 39 74)(31 40 75)(32 41 76)
(1 42 77)(2 78 43)(3 44 79)(4 80 45)(5 46 81)(6 82 47)(7 48 83)(8 84 49)(9 50 85)(10 86 51)(11 52 87)(12 88 53)(13 54 89)(14 90 55)(15 56 91)(16 92 57)(17 58 93)(18 94 59)(19 60 95)(20 96 61)(21 62 65)(22 66 63)(23 64 67)(24 68 33)(25 34 69)(26 70 35)(27 36 71)(28 72 37)(29 38 73)(30 74 39)(31 40 75)(32 76 41)
(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)| (1,42,77)(2,43,78)(3,44,79)(4,45,80)(5,46,81)(6,47,82)(7,48,83)(8,49,84)(9,50,85)(10,51,86)(11,52,87)(12,53,88)(13,54,89)(14,55,90)(15,56,91)(16,57,92)(17,58,93)(18,59,94)(19,60,95)(20,61,96)(21,62,65)(22,63,66)(23,64,67)(24,33,68)(25,34,69)(26,35,70)(27,36,71)(28,37,72)(29,38,73)(30,39,74)(31,40,75)(32,41,76), (1,42,77)(2,78,43)(3,44,79)(4,80,45)(5,46,81)(6,82,47)(7,48,83)(8,84,49)(9,50,85)(10,86,51)(11,52,87)(12,88,53)(13,54,89)(14,90,55)(15,56,91)(16,92,57)(17,58,93)(18,94,59)(19,60,95)(20,96,61)(21,62,65)(22,66,63)(23,64,67)(24,68,33)(25,34,69)(26,70,35)(27,36,71)(28,72,37)(29,38,73)(30,74,39)(31,40,75)(32,76,41), (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( (1,42,77)(2,43,78)(3,44,79)(4,45,80)(5,46,81)(6,47,82)(7,48,83)(8,49,84)(9,50,85)(10,51,86)(11,52,87)(12,53,88)(13,54,89)(14,55,90)(15,56,91)(16,57,92)(17,58,93)(18,59,94)(19,60,95)(20,61,96)(21,62,65)(22,63,66)(23,64,67)(24,33,68)(25,34,69)(26,35,70)(27,36,71)(28,37,72)(29,38,73)(30,39,74)(31,40,75)(32,41,76), (1,42,77)(2,78,43)(3,44,79)(4,80,45)(5,46,81)(6,82,47)(7,48,83)(8,84,49)(9,50,85)(10,86,51)(11,52,87)(12,88,53)(13,54,89)(14,90,55)(15,56,91)(16,92,57)(17,58,93)(18,94,59)(19,60,95)(20,96,61)(21,62,65)(22,66,63)(23,64,67)(24,68,33)(25,34,69)(26,70,35)(27,36,71)(28,72,37)(29,38,73)(30,74,39)(31,40,75)(32,76,41), (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([[(1,42,77),(2,43,78),(3,44,79),(4,45,80),(5,46,81),(6,47,82),(7,48,83),(8,49,84),(9,50,85),(10,51,86),(11,52,87),(12,53,88),(13,54,89),(14,55,90),(15,56,91),(16,57,92),(17,58,93),(18,59,94),(19,60,95),(20,61,96),(21,62,65),(22,63,66),(23,64,67),(24,33,68),(25,34,69),(26,35,70),(27,36,71),(28,37,72),(29,38,73),(30,39,74),(31,40,75),(32,41,76)], [(1,42,77),(2,78,43),(3,44,79),(4,80,45),(5,46,81),(6,82,47),(7,48,83),(8,84,49),(9,50,85),(10,86,51),(11,52,87),(12,88,53),(13,54,89),(14,90,55),(15,56,91),(16,92,57),(17,58,93),(18,94,59),(19,60,95),(20,96,61),(21,62,65),(22,66,63),(23,64,67),(24,68,33),(25,34,69),(26,70,35),(27,36,71),(28,72,37),(29,38,73),(30,74,39),(31,40,75),(32,76,41)], [(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)]])

144 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12J 16A ··· 16H 24A ··· 24H 24I ··· 24T 32A ··· 32P 48A ··· 48P 48Q ··· 48AN 96A ··· 96AF order 1 2 3 3 3 3 3 4 4 6 6 6 6 6 8 8 8 8 12 12 12 12 12 ··· 12 16 ··· 16 24 ··· 24 24 ··· 24 32 ··· 32 48 ··· 48 48 ··· 48 96 ··· 96 size 1 1 1 1 2 2 2 1 1 1 1 2 2 2 1 1 1 1 1 1 1 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + - image C1 C2 C3 C4 C6 C8 C12 C16 C24 C32 C48 C96 S3 Dic3 C3×S3 C3⋊C8 C3×Dic3 C3⋊C16 C3×C3⋊C8 C3⋊C32 C3×C3⋊C16 C3×C3⋊C32 kernel C3×C3⋊C32 C3×C48 C3⋊C32 C3×C24 C48 C3×C12 C24 C3×C6 C12 C32 C6 C3 C48 C24 C16 C12 C8 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 4 4 8 8 16 16 32 1 1 2 2 2 4 4 8 8 16

Matrix representation of C3×C3⋊C32 in GL3(𝔽97) generated by

 35 0 0 0 61 0 0 0 61
,
 1 0 0 0 61 0 0 0 35
,
 55 0 0 0 0 1 0 64 0
G:=sub<GL(3,GF(97))| [35,0,0,0,61,0,0,0,61],[1,0,0,0,61,0,0,0,35],[55,0,0,0,0,64,0,1,0] >;

C3×C3⋊C32 in GAP, Magma, Sage, TeX

C_3\times C_3\rtimes C_{32}
% in TeX

G:=Group("C3xC3:C32");
// GroupNames label

G:=SmallGroup(288,64);
// by ID

G=gap.SmallGroup(288,64);
# by ID

G:=PCGroup([7,-2,-3,-2,-2,-2,-2,-3,42,58,80,102,9414]);
// Polycyclic

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

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