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## G = C62.39D4order 288 = 25·32

### 23rd non-split extension by C62 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C62.39D4
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C12.58D6 — C62.39D4
 Lower central C32 — C3×C6 — C3×C12 — C62.39D4
 Upper central C1 — C4 — C2×C4 — C4○D4

Generators and relations for C62.39D4
G = < a,b,c,d | a6=b6=1, c4=b3, d2=a3b3, ab=ba, cac-1=a-1b3, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a3c3 >

Subgroups: 380 in 132 conjugacy classes, 57 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C32, Dic3, C12, C12, C2×C6, C2×C6, C42, M4(2), C4○D4, C3×C6, C3×C6, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C4≀C2, C3⋊Dic3, C3×C12, C3×C12, C62, C62, C4.Dic3, C4×Dic3, C3×C4○D4, C324C8, C2×C3⋊Dic3, C6×C12, C6×C12, D4×C32, D4×C32, Q8×C32, Q83Dic3, C12.58D6, C4×C3⋊Dic3, C32×C4○D4, C62.39D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, C3⋊S3, C2×Dic3, C3⋊D4, C4≀C2, C3⋊Dic3, C2×C3⋊S3, C6.D4, C2×C3⋊Dic3, C327D4, Q83Dic3, C625C4, C62.39D4

Smallest permutation representation of C62.39D4
On 72 points
Generators in S72
```(1 65 50 5 69 54)(2 51 70)(3 67 52 7 71 56)(4 53 72)(6 55 66)(8 49 68)(9 36 41 13 40 45)(10 42 33)(11 38 43 15 34 47)(12 44 35)(14 46 37)(16 48 39)(17 32 60)(18 57 25 22 61 29)(19 26 62)(20 59 27 24 63 31)(21 28 64)(23 30 58)
(1 34 25 5 38 29)(2 30 39 6 26 35)(3 36 27 7 40 31)(4 32 33 8 28 37)(9 59 52 13 63 56)(10 49 64 14 53 60)(11 61 54 15 57 50)(12 51 58 16 55 62)(17 42 68 21 46 72)(18 65 47 22 69 43)(19 44 70 23 48 66)(20 67 41 24 71 45)
(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 5)(2 8 6 4)(3 7)(9 24)(10 19 14 23)(11 18)(12 21 16 17)(13 20)(15 22)(25 34)(26 37 30 33)(27 36)(28 39 32 35)(29 38)(31 40)(41 59)(42 62 46 58)(43 61)(44 64 48 60)(45 63)(47 57)(49 66 53 70)(50 65)(51 68 55 72)(52 67)(54 69)(56 71)```

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E ··· 6P 8A 8B 12A ··· 12H 12I ··· 12T order 1 2 2 2 3 3 3 3 4 4 4 4 4 4 4 4 6 6 6 6 6 ··· 6 8 8 12 ··· 12 12 ··· 12 size 1 1 2 4 2 2 2 2 1 1 2 4 18 18 18 18 2 2 2 2 4 ··· 4 36 36 2 ··· 2 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 type + + + + + + + + - - image C1 C2 C2 C2 C4 C4 S3 D4 D4 D6 Dic3 Dic3 C3⋊D4 C3⋊D4 C4≀C2 Q8⋊3Dic3 kernel C62.39D4 C12.58D6 C4×C3⋊Dic3 C32×C4○D4 D4×C32 Q8×C32 C3×C4○D4 C3×C12 C62 C2×C12 C3×D4 C3×Q8 C12 C2×C6 C32 C3 # reps 1 1 1 1 2 2 4 1 1 4 4 4 8 8 4 8

Matrix representation of C62.39D4 in GL6(𝔽73)

 72 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 71 70 0 0 0 0 1 1
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 27 0 0 0 0 72 0 0 0 0 0 0 0 43 43 0 0 0 0 13 30 0 0 0 0 0 0 17 34 0 0 0 0 43 56
,
 72 0 0 0 0 0 0 27 0 0 0 0 0 0 1 72 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 72 72

`G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,71,1,0,0,0,0,70,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,72,0,0,0,0,27,0,0,0,0,0,0,0,43,13,0,0,0,0,43,30,0,0,0,0,0,0,17,43,0,0,0,0,34,56],[72,0,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;`

C62.39D4 in GAP, Magma, Sage, TeX

`C_6^2._{39}D_4`
`% in TeX`

`G:=Group("C6^2.39D4");`
`// GroupNames label`

`G:=SmallGroup(288,312);`
`// by ID`

`G=gap.SmallGroup(288,312);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,100,675,346,80,2693,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^6=1,c^4=b^3,d^2=a^3*b^3,a*b=b*a,c*a*c^-1=a^-1*b^3,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^3*c^3>;`
`// generators/relations`

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