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## G = C62.7D4order 288 = 25·32

### 7th non-split extension by C62 of D4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊Dic3 — C62.7D4
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C2×C3⋊Dic3 — C62.C22 — C62.7D4
 Lower central C32 — C3×C6 — C3⋊Dic3 — C62.7D4
 Upper central C1 — C22

Generators and relations for C62.7D4
G = < a,b,c,d | a6=b6=1, c4=b3, d2=a3, ab=ba, cac-1=a3b4, dad-1=a-1, cbc-1=a2b3, bd=db, dcd-1=b3c3 >

Subgroups: 264 in 62 conjugacy classes, 19 normal (15 characteristic)
C1, C2 [×3], C3 [×2], C4 [×4], C22, C6 [×6], C8 [×2], C2×C4 [×3], C32, Dic3 [×6], C12 [×2], C2×C6 [×2], C4⋊C4 [×2], C2×C8, C3×C6 [×3], C2×Dic3 [×4], C2×C12 [×2], C2.D8, C3×Dic3 [×2], C3⋊Dic3 [×2], C62, Dic3⋊C4 [×2], C322C8 [×2], C6×Dic3 [×2], C2×C3⋊Dic3, C62.C22 [×2], C2×C322C8, C62.7D4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4, Q8, C4⋊C4, D8, Q16, C2.D8, S3≀C2, C3⋊S3.Q8, C32⋊D8, C32⋊Q16, C62.7D4

Character table of C62.7D4

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 1 1 4 4 12 12 12 12 18 18 4 4 4 4 4 4 18 18 18 18 12 12 12 12 12 12 12 12 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 1 -1 -1 1 1 1 i -i i -i -1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 -i -i i i i i -i -i linear of order 4 ρ6 1 -1 -1 1 1 1 -i -i i i -1 1 -1 -1 1 -1 1 -1 -1 1 -1 1 i -i -i -i i i -i i linear of order 4 ρ7 1 -1 -1 1 1 1 i i -i -i -1 1 -1 -1 1 -1 1 -1 -1 1 -1 1 -i i i i -i -i i -i linear of order 4 ρ8 1 -1 -1 1 1 1 -i i -i i -1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 i i -i -i -i -i i i linear of order 4 ρ9 2 2 2 2 2 2 0 0 0 0 -2 -2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 2 -2 2 2 0 0 0 0 0 0 -2 2 -2 -2 -2 2 √2 √2 -√2 -√2 0 0 0 0 0 0 0 0 orthogonal lifted from D8 ρ11 2 -2 2 -2 2 2 0 0 0 0 0 0 -2 2 -2 -2 -2 2 -√2 -√2 √2 √2 0 0 0 0 0 0 0 0 orthogonal lifted from D8 ρ12 2 -2 -2 2 2 2 0 0 0 0 2 -2 -2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ13 2 2 -2 -2 2 2 0 0 0 0 0 0 2 -2 -2 2 -2 -2 √2 -√2 -√2 √2 0 0 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ14 2 2 -2 -2 2 2 0 0 0 0 0 0 2 -2 -2 2 -2 -2 -√2 √2 √2 -√2 0 0 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ15 4 4 4 4 1 -2 -2 0 0 -2 0 0 -2 -2 -2 1 1 1 0 0 0 0 1 0 1 1 0 0 0 1 orthogonal lifted from S3≀C2 ρ16 4 4 4 4 -2 1 0 2 2 0 0 0 1 1 1 -2 -2 -2 0 0 0 0 0 -1 0 0 -1 -1 -1 0 orthogonal lifted from S3≀C2 ρ17 4 4 4 4 -2 1 0 -2 -2 0 0 0 1 1 1 -2 -2 -2 0 0 0 0 0 1 0 0 1 1 1 0 orthogonal lifted from S3≀C2 ρ18 4 4 4 4 1 -2 2 0 0 2 0 0 -2 -2 -2 1 1 1 0 0 0 0 -1 0 -1 -1 0 0 0 -1 orthogonal lifted from S3≀C2 ρ19 4 4 -4 -4 -2 1 0 0 0 0 0 0 1 -1 -1 -2 2 2 0 0 0 0 0 -√3 0 0 √3 -√3 √3 0 symplectic lifted from C32⋊Q16, Schur index 2 ρ20 4 4 -4 -4 1 -2 0 0 0 0 0 0 -2 2 2 1 -1 -1 0 0 0 0 √3 0 √3 -√3 0 0 0 -√3 symplectic lifted from C32⋊Q16, Schur index 2 ρ21 4 4 -4 -4 1 -2 0 0 0 0 0 0 -2 2 2 1 -1 -1 0 0 0 0 -√3 0 -√3 √3 0 0 0 √3 symplectic lifted from C32⋊Q16, Schur index 2 ρ22 4 4 -4 -4 -2 1 0 0 0 0 0 0 1 -1 -1 -2 2 2 0 0 0 0 0 √3 0 0 -√3 √3 -√3 0 symplectic lifted from C32⋊Q16, Schur index 2 ρ23 4 -4 4 -4 1 -2 0 0 0 0 0 0 2 -2 2 -1 -1 1 0 0 0 0 √-3 0 -√-3 √-3 0 0 0 -√-3 complex lifted from C32⋊D8 ρ24 4 -4 4 -4 1 -2 0 0 0 0 0 0 2 -2 2 -1 -1 1 0 0 0 0 -√-3 0 √-3 -√-3 0 0 0 √-3 complex lifted from C32⋊D8 ρ25 4 -4 4 -4 -2 1 0 0 0 0 0 0 -1 1 -1 2 2 -2 0 0 0 0 0 -√-3 0 0 -√-3 √-3 √-3 0 complex lifted from C32⋊D8 ρ26 4 -4 4 -4 -2 1 0 0 0 0 0 0 -1 1 -1 2 2 -2 0 0 0 0 0 √-3 0 0 √-3 -√-3 -√-3 0 complex lifted from C32⋊D8 ρ27 4 -4 -4 4 1 -2 2i 0 0 -2i 0 0 2 2 -2 -1 1 -1 0 0 0 0 i 0 -i -i 0 0 0 i complex lifted from C3⋊S3.Q8 ρ28 4 -4 -4 4 -2 1 0 -2i 2i 0 0 0 -1 -1 1 2 -2 2 0 0 0 0 0 i 0 0 -i -i i 0 complex lifted from C3⋊S3.Q8 ρ29 4 -4 -4 4 1 -2 -2i 0 0 2i 0 0 2 2 -2 -1 1 -1 0 0 0 0 -i 0 i i 0 0 0 -i complex lifted from C3⋊S3.Q8 ρ30 4 -4 -4 4 -2 1 0 2i -2i 0 0 0 -1 -1 1 2 -2 2 0 0 0 0 0 -i 0 0 i i -i 0 complex lifted from C3⋊S3.Q8

Smallest permutation representation of C62.7D4
On 96 points
Generators in S96
```(1 51)(2 13 30 52 78 81)(3 53)(4 83 80 54 32 15)(5 55)(6 9 26 56 74 85)(7 49)(8 87 76 50 28 11)(10 75)(12 77)(14 79)(16 73)(17 89 33 60 42 67)(18 61)(19 69 44 62 35 91)(20 63)(21 93 37 64 46 71)(22 57)(23 65 48 58 39 95)(24 59)(25 84)(27 86)(29 88)(31 82)(34 68)(36 70)(38 72)(40 66)(41 96)(43 90)(45 92)(47 94)
(1 73 29 5 77 25)(2 6)(3 27 79 7 31 75)(4 8)(9 13)(10 53 86 14 49 82)(11 15)(12 84 51 16 88 55)(17 21)(18 38 43 22 34 47)(19 23)(20 41 36 24 45 40)(26 30)(28 32)(33 37)(35 39)(42 46)(44 48)(50 54)(52 56)(57 68 94 61 72 90)(58 62)(59 92 66 63 96 70)(60 64)(65 69)(67 71)(74 78)(76 80)(81 85)(83 87)(89 93)(91 95)
(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)
(1 57 51 22)(2 64 52 21)(3 63 53 20)(4 62 54 19)(5 61 55 18)(6 60 56 17)(7 59 49 24)(8 58 50 23)(9 33 74 67)(10 40 75 66)(11 39 76 65)(12 38 77 72)(13 37 78 71)(14 36 79 70)(15 35 80 69)(16 34 73 68)(25 90 84 43)(26 89 85 42)(27 96 86 41)(28 95 87 48)(29 94 88 47)(30 93 81 46)(31 92 82 45)(32 91 83 44)```

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

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

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

Matrix representation of C62.7D4 in GL8(𝔽73)

 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 72 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 45 33 0 0 0 0 0 0 47 28 0 0 0 0 0 0 0 0 0 41 0 0 0 0 0 0 16 41 0 0 0 0 0 0 0 0 0 0 7 14 0 0 0 0 0 0 7 66 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0
,
 26 32 0 0 0 0 0 0 45 47 0 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 7 14 0 0 0 0 0 0 7 66

`G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[45,47,0,0,0,0,0,0,33,28,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,41,41,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,7,7,0,0,0,0,0,0,14,66,0,0],[26,45,0,0,0,0,0,0,32,47,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7,7,0,0,0,0,0,0,14,66] >;`

C62.7D4 in GAP, Magma, Sage, TeX

`C_6^2._7D_4`
`% in TeX`

`G:=Group("C6^2.7D4");`
`// GroupNames label`

`G:=SmallGroup(288,391);`
`// by ID`

`G=gap.SmallGroup(288,391);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,85,64,422,219,100,2693,2028,691,797,2372]);`
`// 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^3*b^4,d*a*d^-1=a^-1,c*b*c^-1=a^2*b^3,b*d=d*b,d*c*d^-1=b^3*c^3>;`
`// generators/relations`

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