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## G = C62.37D4order 288 = 25·32

### 21st non-split extension by C62 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C62.37D4
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C12.59D6 — C62.37D4
 Lower central C32 — C3×C6 — C3×C12 — C62.37D4
 Upper central C1 — C4 — C2×C4 — M4(2)

Generators and relations for C62.37D4
G = < a,b,c,d | a6=b6=1, c4=b3, d2=a3, ab=ba, cac-1=ab3, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=a3b3c3 >

Subgroups: 572 in 132 conjugacy classes, 47 normal (19 characteristic)
C1, C2, C2 [×2], C3 [×4], C4 [×2], C4 [×3], C22, C22, S3 [×4], C6 [×4], C6 [×4], C8, C2×C4, C2×C4 [×2], D4 [×2], Q8, C32, Dic3 [×12], C12 [×8], D6 [×4], C2×C6 [×4], C42, M4(2), C4○D4, C3⋊S3, C3×C6, C3×C6, C24 [×4], Dic6 [×4], C4×S3 [×4], D12 [×4], C2×Dic3 [×4], C3⋊D4 [×4], C2×C12 [×4], C4≀C2, C3⋊Dic3 [×3], C3×C12 [×2], C2×C3⋊S3, C62, C4×Dic3 [×4], C3×M4(2) [×4], C4○D12 [×4], C3×C24, C324Q8, C4×C3⋊S3, C12⋊S3, C2×C3⋊Dic3, C327D4, C6×C12, D12⋊C4 [×4], C4×C3⋊Dic3, C32×M4(2), C12.59D6, C62.37D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4 [×2], D6 [×4], C22⋊C4, C3⋊S3, C4×S3 [×4], D12 [×4], C3⋊D4 [×4], C4≀C2, C2×C3⋊S3, D6⋊C4 [×4], C4×C3⋊S3, C12⋊S3, C327D4, D12⋊C4 [×4], C6.11D12, C62.37D4

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

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

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

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

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 6F 6G 6H 8A 8B 12A ··· 12H 12I 12J 12K 12L 24A ··· 24P order 1 2 2 2 3 3 3 3 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 6 8 8 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 2 36 2 2 2 2 1 1 2 18 18 18 18 36 2 2 2 2 4 4 4 4 4 4 2 ··· 2 4 4 4 4 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C4 C4 S3 D4 D4 D6 C4×S3 C3⋊D4 D12 C4≀C2 D12⋊C4 kernel C62.37D4 C4×C3⋊Dic3 C32×M4(2) C12.59D6 C32⋊4Q8 C12⋊S3 C3×M4(2) C3×C12 C62 C2×C12 C12 C12 C2×C6 C32 C3 # reps 1 1 1 1 2 2 4 1 1 4 8 8 8 4 8

Matrix representation of C62.37D4 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 72 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1
,
 0 72 0 0 0 0 1 72 0 0 0 0 0 0 72 72 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 7 59 0 0 0 0 14 66 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 46 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 72 72 0 0 0 0 0 0 27 0 0 0 0 0 0 72

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[7,14,0,0,0,0,59,66,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,46,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,27,0,0,0,0,0,0,72] >;`

C62.37D4 in GAP, Magma, Sage, TeX

`C_6^2._{37}D_4`
`% in TeX`

`G:=Group("C6^2.37D4");`
`// GroupNames label`

`G:=SmallGroup(288,300);`
`// by ID`

`G=gap.SmallGroup(288,300);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,100,675,346,80,2693,9414]);`
`// Polycyclic`

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

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