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G = C32⋊2Q32order 288 = 25·32

1st semidirect product of C32 and Q32 acting via Q32/C8=C22

Aliases: C322Q32, C24.17D6, Dic12.2S3, C8.15S32, (C3×C6).11D8, C32(C3⋊Q32), (C3×C12).26D4, C6.10(D4⋊S3), C4.3(D6⋊S3), C12.23(C3⋊D4), C24.S3.1C2, (C3×C24).11C22, (C3×Dic12).2C2, C2.5(C322D8), SmallGroup(288,198)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C24 — C32⋊2Q32
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×C24 — C3×Dic12 — C32⋊2Q32
 Lower central C32 — C3×C6 — C3×C12 — C3×C24 — C32⋊2Q32
 Upper central C1 — C2 — C4 — C8

Generators and relations for C322Q32
G = < a,b,c,d | a3=b3=c16=1, d2=c8, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=c-1 >

Smallest permutation representation of C322Q32
On 96 points
Generators in S96
```(1 18 43)(2 44 19)(3 20 45)(4 46 21)(5 22 47)(6 48 23)(7 24 33)(8 34 25)(9 26 35)(10 36 27)(11 28 37)(12 38 29)(13 30 39)(14 40 31)(15 32 41)(16 42 17)(49 70 83)(50 84 71)(51 72 85)(52 86 73)(53 74 87)(54 88 75)(55 76 89)(56 90 77)(57 78 91)(58 92 79)(59 80 93)(60 94 65)(61 66 95)(62 96 67)(63 68 81)(64 82 69)
(1 43 18)(2 19 44)(3 45 20)(4 21 46)(5 47 22)(6 23 48)(7 33 24)(8 25 34)(9 35 26)(10 27 36)(11 37 28)(12 29 38)(13 39 30)(14 31 40)(15 41 32)(16 17 42)(49 70 83)(50 84 71)(51 72 85)(52 86 73)(53 74 87)(54 88 75)(55 76 89)(56 90 77)(57 78 91)(58 92 79)(59 80 93)(60 94 65)(61 66 95)(62 96 67)(63 68 81)(64 82 69)
(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 82 9 90)(2 81 10 89)(3 96 11 88)(4 95 12 87)(5 94 13 86)(6 93 14 85)(7 92 15 84)(8 91 16 83)(17 49 25 57)(18 64 26 56)(19 63 27 55)(20 62 28 54)(21 61 29 53)(22 60 30 52)(23 59 31 51)(24 58 32 50)(33 79 41 71)(34 78 42 70)(35 77 43 69)(36 76 44 68)(37 75 45 67)(38 74 46 66)(39 73 47 65)(40 72 48 80)```

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

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

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

33 conjugacy classes

 class 1 2 3A 3B 3C 4A 4B 4C 6A 6B 6C 8A 8B 12A 12B 12C 12D 12E 12F 12G 12H 16A 16B 16C 16D 24A ··· 24H order 1 2 3 3 3 4 4 4 6 6 6 8 8 12 12 12 12 12 12 12 12 16 16 16 16 24 ··· 24 size 1 1 2 2 4 2 24 24 2 2 4 2 2 4 4 4 4 24 24 24 24 18 18 18 18 4 ··· 4

33 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + - + + - - image C1 C2 C2 S3 D4 D6 D8 C3⋊D4 Q32 S32 D4⋊S3 D6⋊S3 C3⋊Q32 C32⋊2D8 C32⋊2Q32 kernel C32⋊2Q32 C24.S3 C3×Dic12 Dic12 C3×C12 C24 C3×C6 C12 C32 C8 C6 C4 C3 C2 C1 # reps 1 1 2 2 1 2 2 4 4 1 2 1 4 2 4

Matrix representation of C322Q32 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
,
 73 93 0 0 0 0 2 69 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 17 0 0 0 0 57 0 0 0 0 0 0 0 96 0 0 0 0 0 0 96 0 0 0 0 0 0 0 1 0 0 0 0 1 0

`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],[73,2,0,0,0,0,93,69,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,57,0,0,0,0,17,0,0,0,0,0,0,0,96,0,0,0,0,0,0,96,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

C322Q32 in GAP, Magma, Sage, TeX

`C_3^2\rtimes_2Q_{32}`
`% in TeX`

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

`G:=SmallGroup(288,198);`
`// by ID`

`G=gap.SmallGroup(288,198);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,85,120,254,135,142,675,346,80,1356,9414]);`
`// 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=d*a*d^-1=a^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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