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## G = C32⋊3Q32order 288 = 25·32

### 2nd semidirect product of C32 and Q32 acting via Q32/C8=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C24 — C32⋊3Q32
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×C24 — C3×Dic12 — C32⋊3Q32
 Lower central C32 — C3×C6 — C3×C12 — C3×C24 — C32⋊3Q32
 Upper central C1 — C2 — C4 — C8

Generators and relations for C323Q32
G = < a,b,c,d | a3=b3=c16=1, d2=c8, ab=ba, cac-1=a-1, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 258 in 57 conjugacy classes, 22 normal (all characteristic)
C1, C2, C3 [×2], C3, C4, C4 [×2], C6 [×2], C6, C8, Q8 [×2], C32, Dic3 [×5], C12 [×2], C12 [×2], C16, Q16 [×2], C3×C6, C24 [×2], C24, Dic6 [×5], C3×Q8, Q32, C3×Dic3, C3⋊Dic3, C3×C12, C3⋊C16, C48, Dic12, Dic12 [×3], C3×Q16, C3×C24, C3×Dic6, C324Q8, Dic24, C3⋊Q32, C3×C3⋊C16, C3×Dic12, C325Q16, C323Q32
Quotients: C1, C2 [×3], C22, S3 [×2], D4, D6 [×2], D8, D12, C3⋊D4, Q32, S32, D24, D4⋊S3, C3⋊D12, Dic24, C3⋊Q32, C3⋊D24, C323Q32

Smallest permutation representation of C323Q32
On 96 points
Generators in S96
```(1 41 54)(2 55 42)(3 43 56)(4 57 44)(5 45 58)(6 59 46)(7 47 60)(8 61 48)(9 33 62)(10 63 34)(11 35 64)(12 49 36)(13 37 50)(14 51 38)(15 39 52)(16 53 40)(17 68 88)(18 89 69)(19 70 90)(20 91 71)(21 72 92)(22 93 73)(23 74 94)(24 95 75)(25 76 96)(26 81 77)(27 78 82)(28 83 79)(29 80 84)(30 85 65)(31 66 86)(32 87 67)
(1 54 41)(2 55 42)(3 56 43)(4 57 44)(5 58 45)(6 59 46)(7 60 47)(8 61 48)(9 62 33)(10 63 34)(11 64 35)(12 49 36)(13 50 37)(14 51 38)(15 52 39)(16 53 40)(17 68 88)(18 69 89)(19 70 90)(20 71 91)(21 72 92)(22 73 93)(23 74 94)(24 75 95)(25 76 96)(26 77 81)(27 78 82)(28 79 83)(29 80 84)(30 65 85)(31 66 86)(32 67 87)
(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)
(1 25 9 17)(2 24 10 32)(3 23 11 31)(4 22 12 30)(5 21 13 29)(6 20 14 28)(7 19 15 27)(8 18 16 26)(33 68 41 76)(34 67 42 75)(35 66 43 74)(36 65 44 73)(37 80 45 72)(38 79 46 71)(39 78 47 70)(40 77 48 69)(49 85 57 93)(50 84 58 92)(51 83 59 91)(52 82 60 90)(53 81 61 89)(54 96 62 88)(55 95 63 87)(56 94 64 86)```

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

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

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

42 conjugacy classes

 class 1 2 3A 3B 3C 4A 4B 4C 6A 6B 6C 8A 8B 12A 12B 12C 12D 12E 12F 12G 16A 16B 16C 16D 24A 24B 24C 24D 24E ··· 24J 48A ··· 48H order 1 2 3 3 3 4 4 4 6 6 6 8 8 12 12 12 12 12 12 12 16 16 16 16 24 24 24 24 24 ··· 24 48 ··· 48 size 1 1 2 2 4 2 24 72 2 2 4 2 2 2 2 4 4 4 24 24 6 6 6 6 2 2 2 2 4 ··· 4 6 ··· 6

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + - + - + + + - + - image C1 C2 C2 C2 S3 S3 D4 D6 D8 D12 C3⋊D4 Q32 D24 Dic24 S32 D4⋊S3 C3⋊D12 C3⋊Q32 C3⋊D24 C32⋊3Q32 kernel C32⋊3Q32 C3×C3⋊C16 C3×Dic12 C32⋊5Q16 C3⋊C16 Dic12 C3×C12 C24 C3×C6 C12 C12 C32 C6 C3 C8 C6 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 2 2 2 2 4 4 8 1 1 1 2 2 4

Matrix representation of C323Q32 in GL6(𝔽97)

 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 0 96 0 0 0 0 1 96
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 96 0 0 0 0 1 96 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 24 4 0 0 0 0 95 28 0 0 0 0 0 0 29 39 0 0 0 0 58 68 0 0 0 0 0 0 82 56 0 0 0 0 41 15
,
 95 75 0 0 0 0 84 2 0 0 0 0 0 0 39 29 0 0 0 0 68 58 0 0 0 0 0 0 41 15 0 0 0 0 82 56

`G:=sub<GL(6,GF(97))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,96,96],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,96,96,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[24,95,0,0,0,0,4,28,0,0,0,0,0,0,29,58,0,0,0,0,39,68,0,0,0,0,0,0,82,41,0,0,0,0,56,15],[95,84,0,0,0,0,75,2,0,0,0,0,0,0,39,68,0,0,0,0,29,58,0,0,0,0,0,0,41,82,0,0,0,0,15,56] >;`

C323Q32 in GAP, Magma, Sage, TeX

`C_3^2\rtimes_3Q_{32}`
`% in TeX`

`G:=Group("C3^2:3Q32");`
`// GroupNames label`

`G:=SmallGroup(288,199);`
`// by ID`

`G=gap.SmallGroup(288,199);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,85,92,254,142,675,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^3=b^3=c^16=1,d^2=c^8,a*b=b*a,c*a*c^-1=a^-1,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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