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G = C3⋊D32order 192 = 26·3

The semidirect product of C3 and D32 acting via D32/D16=C2

Aliases: C32D32, D483C2, D161S3, C12.5D8, C6.8D16, C24.9D4, C16.4D6, C48.2C22, C3⋊C321C2, (C3×D16)⋊1C2, C4.1(D4⋊S3), C8.9(C3⋊D4), C2.4(C3⋊D16), SmallGroup(192,78)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C48 — C3⋊D32
 Chief series C1 — C3 — C6 — C12 — C24 — C48 — D48 — C3⋊D32
 Lower central C3 — C6 — C12 — C24 — C48 — C3⋊D32
 Upper central C1 — C2 — C4 — C8 — C16 — D16

Generators and relations for C3⋊D32
G = < a,b,c | a3=b32=c2=1, bab-1=cac=a-1, cbc=b-1 >

Character table of C3⋊D32

 class 1 2A 2B 2C 3 4 6A 6B 6C 8A 8B 12 16A 16B 16C 16D 24A 24B 32A 32B 32C 32D 32E 32F 32G 32H 48A 48B 48C 48D size 1 1 16 48 2 2 2 16 16 2 2 4 2 2 2 2 4 4 6 6 6 6 6 6 6 6 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ3 1 1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ4 1 1 -1 -1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 2 2 2 0 -1 2 -1 -1 -1 2 2 -1 2 2 2 2 -1 -1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 2 0 0 2 2 2 0 0 2 2 2 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 -2 -2 -2 -2 orthogonal lifted from D4 ρ7 2 2 -2 0 -1 2 -1 1 1 2 2 -1 2 2 2 2 -1 -1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from D6 ρ8 2 2 0 0 2 -2 2 0 0 0 0 -2 √2 -√2 √2 -√2 0 0 -ζ167+ζ16 -ζ165+ζ163 ζ165-ζ163 ζ165-ζ163 ζ167-ζ16 ζ167-ζ16 -ζ167+ζ16 -ζ165+ζ163 -√2 √2 -√2 √2 orthogonal lifted from D16 ρ9 2 2 0 0 2 -2 2 0 0 0 0 -2 √2 -√2 √2 -√2 0 0 ζ167-ζ16 ζ165-ζ163 -ζ165+ζ163 -ζ165+ζ163 -ζ167+ζ16 -ζ167+ζ16 ζ167-ζ16 ζ165-ζ163 -√2 √2 -√2 √2 orthogonal lifted from D16 ρ10 2 2 0 0 2 2 2 0 0 -2 -2 2 0 0 0 0 -2 -2 -√2 √2 √2 √2 -√2 -√2 -√2 √2 0 0 0 0 orthogonal lifted from D8 ρ11 2 2 0 0 2 2 2 0 0 -2 -2 2 0 0 0 0 -2 -2 √2 -√2 -√2 -√2 √2 √2 √2 -√2 0 0 0 0 orthogonal lifted from D8 ρ12 2 -2 0 0 2 0 -2 0 0 √2 -√2 0 ζ3210-ζ326 -ζ3230+ζ3218 -ζ3210+ζ326 ζ3230-ζ3218 √2 -√2 ζ3211-ζ325 ζ3215-ζ32 -ζ3225+ζ3223 ζ3225-ζ3223 -ζ3213+ζ323 ζ3213-ζ323 -ζ3211+ζ325 -ζ3215+ζ32 ζ3230-ζ3218 ζ3210-ζ326 -ζ3230+ζ3218 -ζ3210+ζ326 orthogonal lifted from D32 ρ13 2 -2 0 0 2 0 -2 0 0 √2 -√2 0 -ζ3210+ζ326 ζ3230-ζ3218 ζ3210-ζ326 -ζ3230+ζ3218 √2 -√2 ζ3213-ζ323 -ζ3225+ζ3223 -ζ3215+ζ32 ζ3215-ζ32 ζ3211-ζ325 -ζ3211+ζ325 -ζ3213+ζ323 ζ3225-ζ3223 -ζ3230+ζ3218 -ζ3210+ζ326 ζ3230-ζ3218 ζ3210-ζ326 orthogonal lifted from D32 ρ14 2 2 0 0 2 -2 2 0 0 0 0 -2 -√2 √2 -√2 √2 0 0 -ζ165+ζ163 ζ167-ζ16 -ζ167+ζ16 -ζ167+ζ16 ζ165-ζ163 ζ165-ζ163 -ζ165+ζ163 ζ167-ζ16 √2 -√2 √2 -√2 orthogonal lifted from D16 ρ15 2 2 0 0 2 -2 2 0 0 0 0 -2 -√2 √2 -√2 √2 0 0 ζ165-ζ163 -ζ167+ζ16 ζ167-ζ16 ζ167-ζ16 -ζ165+ζ163 -ζ165+ζ163 ζ165-ζ163 -ζ167+ζ16 √2 -√2 √2 -√2 orthogonal lifted from D16 ρ16 2 -2 0 0 2 0 -2 0 0 √2 -√2 0 -ζ3210+ζ326 ζ3230-ζ3218 ζ3210-ζ326 -ζ3230+ζ3218 √2 -√2 -ζ3213+ζ323 ζ3225-ζ3223 ζ3215-ζ32 -ζ3215+ζ32 -ζ3211+ζ325 ζ3211-ζ325 ζ3213-ζ323 -ζ3225+ζ3223 -ζ3230+ζ3218 -ζ3210+ζ326 ζ3230-ζ3218 ζ3210-ζ326 orthogonal lifted from D32 ρ17 2 -2 0 0 2 0 -2 0 0 √2 -√2 0 ζ3210-ζ326 -ζ3230+ζ3218 -ζ3210+ζ326 ζ3230-ζ3218 √2 -√2 -ζ3211+ζ325 -ζ3215+ζ32 ζ3225-ζ3223 -ζ3225+ζ3223 ζ3213-ζ323 -ζ3213+ζ323 ζ3211-ζ325 ζ3215-ζ32 ζ3230-ζ3218 ζ3210-ζ326 -ζ3230+ζ3218 -ζ3210+ζ326 orthogonal lifted from D32 ρ18 2 -2 0 0 2 0 -2 0 0 -√2 √2 0 -ζ3230+ζ3218 -ζ3210+ζ326 ζ3230-ζ3218 ζ3210-ζ326 -√2 √2 -ζ3225+ζ3223 ζ3211-ζ325 ζ3213-ζ323 -ζ3213+ζ323 ζ3215-ζ32 -ζ3215+ζ32 ζ3225-ζ3223 -ζ3211+ζ325 ζ3210-ζ326 -ζ3230+ζ3218 -ζ3210+ζ326 ζ3230-ζ3218 orthogonal lifted from D32 ρ19 2 -2 0 0 2 0 -2 0 0 -√2 √2 0 ζ3230-ζ3218 ζ3210-ζ326 -ζ3230+ζ3218 -ζ3210+ζ326 -√2 √2 ζ3215-ζ32 -ζ3213+ζ323 ζ3211-ζ325 -ζ3211+ζ325 ζ3225-ζ3223 -ζ3225+ζ3223 -ζ3215+ζ32 ζ3213-ζ323 -ζ3210+ζ326 ζ3230-ζ3218 ζ3210-ζ326 -ζ3230+ζ3218 orthogonal lifted from D32 ρ20 2 -2 0 0 2 0 -2 0 0 -√2 √2 0 -ζ3230+ζ3218 -ζ3210+ζ326 ζ3230-ζ3218 ζ3210-ζ326 -√2 √2 ζ3225-ζ3223 -ζ3211+ζ325 -ζ3213+ζ323 ζ3213-ζ323 -ζ3215+ζ32 ζ3215-ζ32 -ζ3225+ζ3223 ζ3211-ζ325 ζ3210-ζ326 -ζ3230+ζ3218 -ζ3210+ζ326 ζ3230-ζ3218 orthogonal lifted from D32 ρ21 2 -2 0 0 2 0 -2 0 0 -√2 √2 0 ζ3230-ζ3218 ζ3210-ζ326 -ζ3230+ζ3218 -ζ3210+ζ326 -√2 √2 -ζ3215+ζ32 ζ3213-ζ323 -ζ3211+ζ325 ζ3211-ζ325 -ζ3225+ζ3223 ζ3225-ζ3223 ζ3215-ζ32 -ζ3213+ζ323 -ζ3210+ζ326 ζ3230-ζ3218 ζ3210-ζ326 -ζ3230+ζ3218 orthogonal lifted from D32 ρ22 2 2 0 0 -1 2 -1 √-3 -√-3 2 2 -1 -2 -2 -2 -2 -1 -1 0 0 0 0 0 0 0 0 1 1 1 1 complex lifted from C3⋊D4 ρ23 2 2 0 0 -1 2 -1 -√-3 √-3 2 2 -1 -2 -2 -2 -2 -1 -1 0 0 0 0 0 0 0 0 1 1 1 1 complex lifted from C3⋊D4 ρ24 4 4 0 0 -2 4 -2 0 0 -4 -4 -2 0 0 0 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ25 4 4 0 0 -2 -4 -2 0 0 0 0 2 2√2 -2√2 2√2 -2√2 0 0 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 orthogonal lifted from C3⋊D16, Schur index 2 ρ26 4 4 0 0 -2 -4 -2 0 0 0 0 2 -2√2 2√2 -2√2 2√2 0 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 orthogonal lifted from C3⋊D16, Schur index 2 ρ27 4 -4 0 0 -2 0 2 0 0 2√2 -2√2 0 2ζ165-2ζ163 -2ζ1615+2ζ169 -2ζ165+2ζ163 2ζ1615-2ζ169 -√2 √2 0 0 0 0 0 0 0 0 ζ167-ζ16 -ζ165+ζ163 -ζ167+ζ16 ζ165-ζ163 orthogonal faithful, Schur index 2 ρ28 4 -4 0 0 -2 0 2 0 0 -2√2 2√2 0 -2ζ1615+2ζ169 -2ζ165+2ζ163 2ζ1615-2ζ169 2ζ165-2ζ163 √2 -√2 0 0 0 0 0 0 0 0 -ζ165+ζ163 -ζ167+ζ16 ζ165-ζ163 ζ167-ζ16 orthogonal faithful, Schur index 2 ρ29 4 -4 0 0 -2 0 2 0 0 2√2 -2√2 0 -2ζ165+2ζ163 2ζ1615-2ζ169 2ζ165-2ζ163 -2ζ1615+2ζ169 -√2 √2 0 0 0 0 0 0 0 0 -ζ167+ζ16 ζ165-ζ163 ζ167-ζ16 -ζ165+ζ163 orthogonal faithful, Schur index 2 ρ30 4 -4 0 0 -2 0 2 0 0 -2√2 2√2 0 2ζ1615-2ζ169 2ζ165-2ζ163 -2ζ1615+2ζ169 -2ζ165+2ζ163 √2 -√2 0 0 0 0 0 0 0 0 ζ165-ζ163 ζ167-ζ16 -ζ165+ζ163 -ζ167+ζ16 orthogonal faithful, Schur index 2

Smallest permutation representation of C3⋊D32
On 96 points
Generators in S96
```(1 78 49)(2 50 79)(3 80 51)(4 52 81)(5 82 53)(6 54 83)(7 84 55)(8 56 85)(9 86 57)(10 58 87)(11 88 59)(12 60 89)(13 90 61)(14 62 91)(15 92 63)(16 64 93)(17 94 33)(18 34 95)(19 96 35)(20 36 65)(21 66 37)(22 38 67)(23 68 39)(24 40 69)(25 70 41)(26 42 71)(27 72 43)(28 44 73)(29 74 45)(30 46 75)(31 76 47)(32 48 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)
(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(33 94)(34 93)(35 92)(36 91)(37 90)(38 89)(39 88)(40 87)(41 86)(42 85)(43 84)(44 83)(45 82)(46 81)(47 80)(48 79)(49 78)(50 77)(51 76)(52 75)(53 74)(54 73)(55 72)(56 71)(57 70)(58 69)(59 68)(60 67)(61 66)(62 65)(63 96)(64 95)```

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

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

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

Matrix representation of C3⋊D32 in GL4(𝔽97) generated by

 1 0 0 0 0 1 0 0 0 0 96 1 0 0 96 0
,
 67 59 0 0 74 84 0 0 0 0 82 56 0 0 41 15
,
 1 0 0 0 94 96 0 0 0 0 0 1 0 0 1 0
`G:=sub<GL(4,GF(97))| [1,0,0,0,0,1,0,0,0,0,96,96,0,0,1,0],[67,74,0,0,59,84,0,0,0,0,82,41,0,0,56,15],[1,94,0,0,0,96,0,0,0,0,0,1,0,0,1,0] >;`

C3⋊D32 in GAP, Magma, Sage, TeX

`C_3\rtimes D_{32}`
`% in TeX`

`G:=Group("C3:D32");`
`// GroupNames label`

`G:=SmallGroup(192,78);`
`// by ID`

`G=gap.SmallGroup(192,78);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,85,254,135,142,675,346,192,1684,851,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^3=b^32=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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