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## G = C62.131D4order 288 = 25·32

### 36th non-split extension by C62 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C62.131D4
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C12⋊S3 — C12.59D6 — C62.131D4
 Lower central C32 — C3×C6 — C3×C12 — C62.131D4
 Upper central C1 — C2 — C2×C4 — C2×D4

Generators and relations for C62.131D4
G = < a,b,c,d | a6=b6=d2=1, c4=b3, ab=ba, cac-1=dad=a-1b3, cbc-1=dbd=b-1, dcd=b3c3 >

Subgroups: 716 in 204 conjugacy classes, 65 normal (19 characteristic)
C1, C2, C2 [×4], C3 [×4], C4 [×2], C4, C22, C22 [×5], S3 [×4], C6 [×4], C6 [×12], C8 [×2], C2×C4, C2×C4, D4 [×2], D4 [×3], Q8, C23, C32, Dic3 [×4], C12 [×8], D6 [×4], C2×C6 [×4], C2×C6 [×16], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3⋊S3, C3×C6, C3×C6 [×3], C3⋊C8 [×8], Dic6 [×4], C4×S3 [×4], D12 [×4], C3⋊D4 [×4], C2×C12 [×4], C3×D4 [×8], C3×D4 [×4], C22×C6 [×4], C8⋊C22, C3⋊Dic3, C3×C12 [×2], C2×C3⋊S3, C62, C62 [×4], C4.Dic3 [×4], D4⋊S3 [×8], D4.S3 [×8], C4○D12 [×4], C6×D4 [×4], C324C8 [×2], C324Q8, C4×C3⋊S3, C12⋊S3, C327D4, C6×C12, D4×C32 [×2], D4×C32, C2×C62, D126C22 [×4], C12.58D6, C327D8 [×2], C329SD16 [×2], C12.59D6, D4×C3×C6, C62.131D4
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C3⋊S3, C3⋊D4 [×8], C22×S3 [×4], C8⋊C22, C2×C3⋊S3 [×3], C2×C3⋊D4 [×4], C327D4 [×2], C22×C3⋊S3, D126C22 [×4], C2×C327D4, C62.131D4

Smallest permutation representation of C62.131D4
On 72 points
Generators in S72
```(1 25 68 5 29 72)(2 69 30)(3 27 70 7 31 66)(4 71 32)(6 65 26)(8 67 28)(9 33 57 13 37 61)(10 58 38)(11 35 59 15 39 63)(12 60 40)(14 62 34)(16 64 36)(17 48 52)(18 49 41 22 53 45)(19 42 54)(20 51 43 24 55 47)(21 44 56)(23 46 50)
(1 18 15 5 22 11)(2 12 23 6 16 19)(3 20 9 7 24 13)(4 14 17 8 10 21)(25 49 39 29 53 35)(26 36 54 30 40 50)(27 51 33 31 55 37)(28 38 56 32 34 52)(41 63 72 45 59 68)(42 69 60 46 65 64)(43 57 66 47 61 70)(44 71 62 48 67 58)
(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 8)(2 7)(3 6)(4 5)(9 12)(10 11)(13 16)(14 15)(17 18)(19 24)(20 23)(21 22)(25 32)(26 31)(27 30)(28 29)(33 36)(34 35)(37 40)(38 39)(41 48)(42 47)(43 46)(44 45)(49 56)(50 55)(51 54)(52 53)(57 60)(58 59)(61 64)(62 63)(65 70)(66 69)(67 68)(71 72)```

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

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

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

51 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 6A ··· 6L 6M ··· 6AB 8A 8B 12A ··· 12H order 1 2 2 2 2 2 3 3 3 3 4 4 4 6 ··· 6 6 ··· 6 8 8 12 ··· 12 size 1 1 2 4 4 36 2 2 2 2 2 2 36 2 ··· 2 4 ··· 4 36 36 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 C3⋊D4 C3⋊D4 C8⋊C22 D12⋊6C22 kernel C62.131D4 C12.58D6 C32⋊7D8 C32⋊9SD16 C12.59D6 D4×C3×C6 C6×D4 C3×C12 C62 C2×C12 C3×D4 C12 C2×C6 C32 C3 # reps 1 1 2 2 1 1 4 1 1 4 8 8 8 1 8

Matrix representation of C62.131D4 in GL6(𝔽73)

 9 0 0 0 0 0 0 65 0 0 0 0 0 0 65 0 0 0 0 0 0 65 0 0 0 0 0 0 64 0 0 0 0 0 0 64
,
 64 0 0 0 0 0 0 8 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 0 72 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 72 72 0 0 1 2 0 0 0 0 0 72 0 0
,
 0 72 0 0 0 0 72 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0

`G:=sub<GL(6,GF(73))| [9,0,0,0,0,0,0,65,0,0,0,0,0,0,65,0,0,0,0,0,0,65,0,0,0,0,0,0,64,0,0,0,0,0,0,64],[64,0,0,0,0,0,0,8,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,72,0,0,1,72,0,0,0,0,0,72,0,0],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C62.131D4 in GAP, Magma, Sage, TeX

`C_6^2._{131}D_4`
`% in TeX`

`G:=Group("C6^2.131D4");`
`// GroupNames label`

`G:=SmallGroup(288,789);`
`// by ID`

`G=gap.SmallGroup(288,789);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,675,185,80,2693,9414]);`
`// Polycyclic`

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

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