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## G = C62.4D4order 288 = 25·32

### 4th non-split extension by C62 of D4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊Dic3 — C62.4D4
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C2×C3⋊Dic3 — C62.C22 — C62.4D4
 Lower central C32 — C3×C6 — C3⋊Dic3 — C62.4D4
 Upper central C1 — C22

Generators and relations for C62.4D4
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=a3c3 >

Subgroups: 328 in 74 conjugacy classes, 19 normal (17 characteristic)
C1, C2 [×3], C3 [×2], C4 [×5], C22, C6 [×6], C8, C2×C4 [×3], Q8 [×3], C32, Dic3 [×7], C12 [×3], C2×C6 [×2], C4⋊C4, C2×C8, C2×Q8, C3×C6 [×3], Dic6 [×4], C2×Dic3 [×4], C2×C12 [×2], Q8⋊C4, C3×Dic3 [×3], C3⋊Dic3 [×2], C62, Dic3⋊C4, C2×Dic6, C322C8, C322Q8 [×2], C322Q8, C6×Dic3 [×2], C2×C3⋊Dic3, C62.C22, C2×C322C8, C2×C322Q8, C62.4D4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, SD16, Q16, Q8⋊C4, S3≀C2, S32⋊C4, C322SD16, C32⋊Q16, C62.4D4

Character table of C62.4D4

 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 -1 1 i -1 1 -1 -1 1 -1 1 -1 i -i i -i i -1 -i -i 1 1 -1 i linear of order 4 ρ6 1 -1 -1 1 1 1 -i 1 -1 i -1 1 -1 -1 1 -1 1 -1 -i i -i i i 1 -i -i -1 -1 1 i linear of order 4 ρ7 1 -1 -1 1 1 1 i 1 -1 -i -1 1 -1 -1 1 -1 1 -1 i -i i -i -i 1 i i -1 -1 1 -i linear of order 4 ρ8 1 -1 -1 1 1 1 i -1 1 -i -1 1 -1 -1 1 -1 1 -1 -i i -i i -i -1 i i 1 1 -1 -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 -2 -2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 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 symplectic lifted from Q16, Schur index 2 ρ12 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 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 complex lifted from SD16 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 complex lifted from SD16 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 -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 S32⋊C4 ρ19 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 ρ20 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 S32⋊C4 ρ21 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 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⋊2SD16, 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 symplectic lifted from C32⋊Q16, Schur index 2 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 symplectic lifted from C32⋊Q16, Schur index 2 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 symplectic lifted from C32⋊Q16, Schur index 2 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 symplectic lifted from C32⋊2SD16, Schur index 2 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 S32⋊C4 ρ28 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 S32⋊C4 ρ29 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⋊2SD16 ρ30 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⋊2SD16

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

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

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

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

Matrix representation of C62.4D4 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 1 0 0 0 0 72 0
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 1 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 40 18 0 0 0 0 67 21 0 0 0 0 0 0 0 0 0 27 0 0 0 0 27 0 0 0 7 14 0 0 0 0 59 66 0 0
,
 56 55 0 0 0 0 16 17 0 0 0 0 0 0 66 59 0 0 0 0 14 7 0 0 0 0 0 0 0 27 0 0 0 0 27 0

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,1,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[40,67,0,0,0,0,18,21,0,0,0,0,0,0,0,0,7,59,0,0,0,0,14,66,0,0,0,27,0,0,0,0,27,0,0,0],[56,16,0,0,0,0,55,17,0,0,0,0,0,0,66,14,0,0,0,0,59,7,0,0,0,0,0,0,0,27,0,0,0,0,27,0] >;`

C62.4D4 in GAP, Magma, Sage, TeX

`C_6^2._4D_4`
`% in TeX`

`G:=Group("C6^2.4D4");`
`// GroupNames label`

`G:=SmallGroup(288,388);`
`// by ID`

`G=gap.SmallGroup(288,388);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,85,120,422,219,100,2693,2028,691,797,2372]);`
`// 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=a^3*c^3>;`
`// generators/relations`

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