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## G = C33⋊4Q8order 216 = 23·33

### 2nd semidirect product of C33 and Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C33⋊4Q8
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C32×Dic3 — C33⋊4Q8
 Lower central C33 — C32×C6 — C33⋊4Q8
 Upper central C1 — C2

Generators and relations for C334Q8
G = < a,b,c,d,e | a3=b3=c3=d4=1, e2=d2, ab=ba, ac=ca, dad-1=a-1, ae=ea, bc=cb, dbd-1=b-1, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 396 in 96 conjugacy classes, 34 normal (14 characteristic)
C1, C2, C3, C3 [×4], C3 [×4], C4 [×3], C6, C6 [×4], C6 [×4], Q8, C32, C32 [×4], C32 [×4], Dic3, Dic3 [×13], C12 [×5], C3×C6, C3×C6 [×4], C3×C6 [×4], Dic6 [×5], C33, C3×Dic3 [×4], C3×Dic3 [×4], C3⋊Dic3, C3⋊Dic3 [×9], C3×C12, C32×C6, C322Q8 [×4], C324Q8, C32×Dic3, C3×C3⋊Dic3, C335C4, C334Q8
Quotients: C1, C2 [×3], C22, S3 [×5], Q8, D6 [×5], C3⋊S3, Dic6 [×5], S32 [×4], C2×C3⋊S3, C322Q8 [×4], C324Q8, S3×C3⋊S3, C334Q8

Smallest permutation representation of C334Q8
On 72 points
Generators in S72
```(1 15 33)(2 34 16)(3 13 35)(4 36 14)(5 31 60)(6 57 32)(7 29 58)(8 59 30)(9 54 63)(10 64 55)(11 56 61)(12 62 53)(17 72 65)(18 66 69)(19 70 67)(20 68 71)(21 40 49)(22 50 37)(23 38 51)(24 52 39)(25 41 48)(26 45 42)(27 43 46)(28 47 44)
(1 40 7)(2 8 37)(3 38 5)(4 6 39)(9 46 66)(10 67 47)(11 48 68)(12 65 45)(13 51 31)(14 32 52)(15 49 29)(16 30 50)(17 42 62)(18 63 43)(19 44 64)(20 61 41)(21 58 33)(22 34 59)(23 60 35)(24 36 57)(25 71 56)(26 53 72)(27 69 54)(28 55 70)
(1 21 29)(2 22 30)(3 23 31)(4 24 32)(5 35 51)(6 36 52)(7 33 49)(8 34 50)(9 69 43)(10 70 44)(11 71 41)(12 72 42)(13 38 60)(14 39 57)(15 40 58)(16 37 59)(17 45 53)(18 46 54)(19 47 55)(20 48 56)(25 61 68)(26 62 65)(27 63 66)(28 64 67)
(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)
(1 18 3 20)(2 17 4 19)(5 41 7 43)(6 44 8 42)(9 51 11 49)(10 50 12 52)(13 68 15 66)(14 67 16 65)(21 54 23 56)(22 53 24 55)(25 58 27 60)(26 57 28 59)(29 46 31 48)(30 45 32 47)(33 69 35 71)(34 72 36 70)(37 62 39 64)(38 61 40 63)```

`G:=sub<Sym(72)| (1,15,33)(2,34,16)(3,13,35)(4,36,14)(5,31,60)(6,57,32)(7,29,58)(8,59,30)(9,54,63)(10,64,55)(11,56,61)(12,62,53)(17,72,65)(18,66,69)(19,70,67)(20,68,71)(21,40,49)(22,50,37)(23,38,51)(24,52,39)(25,41,48)(26,45,42)(27,43,46)(28,47,44), (1,40,7)(2,8,37)(3,38,5)(4,6,39)(9,46,66)(10,67,47)(11,48,68)(12,65,45)(13,51,31)(14,32,52)(15,49,29)(16,30,50)(17,42,62)(18,63,43)(19,44,64)(20,61,41)(21,58,33)(22,34,59)(23,60,35)(24,36,57)(25,71,56)(26,53,72)(27,69,54)(28,55,70), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,35,51)(6,36,52)(7,33,49)(8,34,50)(9,69,43)(10,70,44)(11,71,41)(12,72,42)(13,38,60)(14,39,57)(15,40,58)(16,37,59)(17,45,53)(18,46,54)(19,47,55)(20,48,56)(25,61,68)(26,62,65)(27,63,66)(28,64,67), (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), (1,18,3,20)(2,17,4,19)(5,41,7,43)(6,44,8,42)(9,51,11,49)(10,50,12,52)(13,68,15,66)(14,67,16,65)(21,54,23,56)(22,53,24,55)(25,58,27,60)(26,57,28,59)(29,46,31,48)(30,45,32,47)(33,69,35,71)(34,72,36,70)(37,62,39,64)(38,61,40,63)>;`

`G:=Group( (1,15,33)(2,34,16)(3,13,35)(4,36,14)(5,31,60)(6,57,32)(7,29,58)(8,59,30)(9,54,63)(10,64,55)(11,56,61)(12,62,53)(17,72,65)(18,66,69)(19,70,67)(20,68,71)(21,40,49)(22,50,37)(23,38,51)(24,52,39)(25,41,48)(26,45,42)(27,43,46)(28,47,44), (1,40,7)(2,8,37)(3,38,5)(4,6,39)(9,46,66)(10,67,47)(11,48,68)(12,65,45)(13,51,31)(14,32,52)(15,49,29)(16,30,50)(17,42,62)(18,63,43)(19,44,64)(20,61,41)(21,58,33)(22,34,59)(23,60,35)(24,36,57)(25,71,56)(26,53,72)(27,69,54)(28,55,70), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,35,51)(6,36,52)(7,33,49)(8,34,50)(9,69,43)(10,70,44)(11,71,41)(12,72,42)(13,38,60)(14,39,57)(15,40,58)(16,37,59)(17,45,53)(18,46,54)(19,47,55)(20,48,56)(25,61,68)(26,62,65)(27,63,66)(28,64,67), (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), (1,18,3,20)(2,17,4,19)(5,41,7,43)(6,44,8,42)(9,51,11,49)(10,50,12,52)(13,68,15,66)(14,67,16,65)(21,54,23,56)(22,53,24,55)(25,58,27,60)(26,57,28,59)(29,46,31,48)(30,45,32,47)(33,69,35,71)(34,72,36,70)(37,62,39,64)(38,61,40,63) );`

`G=PermutationGroup([(1,15,33),(2,34,16),(3,13,35),(4,36,14),(5,31,60),(6,57,32),(7,29,58),(8,59,30),(9,54,63),(10,64,55),(11,56,61),(12,62,53),(17,72,65),(18,66,69),(19,70,67),(20,68,71),(21,40,49),(22,50,37),(23,38,51),(24,52,39),(25,41,48),(26,45,42),(27,43,46),(28,47,44)], [(1,40,7),(2,8,37),(3,38,5),(4,6,39),(9,46,66),(10,67,47),(11,48,68),(12,65,45),(13,51,31),(14,32,52),(15,49,29),(16,30,50),(17,42,62),(18,63,43),(19,44,64),(20,61,41),(21,58,33),(22,34,59),(23,60,35),(24,36,57),(25,71,56),(26,53,72),(27,69,54),(28,55,70)], [(1,21,29),(2,22,30),(3,23,31),(4,24,32),(5,35,51),(6,36,52),(7,33,49),(8,34,50),(9,69,43),(10,70,44),(11,71,41),(12,72,42),(13,38,60),(14,39,57),(15,40,58),(16,37,59),(17,45,53),(18,46,54),(19,47,55),(20,48,56),(25,61,68),(26,62,65),(27,63,66),(28,64,67)], [(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)], [(1,18,3,20),(2,17,4,19),(5,41,7,43),(6,44,8,42),(9,51,11,49),(10,50,12,52),(13,68,15,66),(14,67,16,65),(21,54,23,56),(22,53,24,55),(25,58,27,60),(26,57,28,59),(29,46,31,48),(30,45,32,47),(33,69,35,71),(34,72,36,70),(37,62,39,64),(38,61,40,63)])`

C334Q8 is a maximal subgroup of
S3×C322Q8  C335(C2×Q8)  D6.S32  D6.4S32  S3×C324Q8  C3⋊S3×Dic6  C329(S3×Q8)  C12.73S32  C62.90D6  C62.91D6  C62.93D6
C334Q8 is a maximal quotient of
C62.80D6  C62.81D6  C62.82D6

33 conjugacy classes

 class 1 2 3A ··· 3E 3F 3G 3H 3I 4A 4B 4C 6A ··· 6E 6F 6G 6H 6I 12A ··· 12H 12I 12J order 1 2 3 ··· 3 3 3 3 3 4 4 4 6 ··· 6 6 6 6 6 12 ··· 12 12 12 size 1 1 2 ··· 2 4 4 4 4 6 18 54 2 ··· 2 4 4 4 4 6 ··· 6 18 18

33 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + - + - + - image C1 C2 C2 C2 S3 S3 Q8 D6 Dic6 S32 C32⋊2Q8 kernel C33⋊4Q8 C32×Dic3 C3×C3⋊Dic3 C33⋊5C4 C3×Dic3 C3⋊Dic3 C33 C3×C6 C32 C6 C3 # reps 1 1 1 1 4 1 1 5 10 4 4

Matrix representation of C334Q8 in GL6(𝔽13)

 3 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 1 12
,
 3 0 0 0 0 0 0 9 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 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 5 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[3,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;`

C334Q8 in GAP, Magma, Sage, TeX

`C_3^3\rtimes_4Q_8`
`% in TeX`

`G:=Group("C3^3:4Q8");`
`// GroupNames label`

`G:=SmallGroup(216,130);`
`// by ID`

`G=gap.SmallGroup(216,130);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,73,31,201,730,5189]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;`
`// generators/relations`

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