Copied to
clipboard

## G = C62.6D4order 288 = 25·32

### 6th non-split extension by C62 of D4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊Dic3 — C62.6D4
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C2×C3⋊Dic3 — C62.C22 — C62.6D4
 Lower central C32 — C3×C6 — C3⋊Dic3 — C62.6D4
 Upper central C1 — C22

Generators and relations for C62.6D4
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=c3 >

Subgroups: 264 in 62 conjugacy classes, 19 normal (11 characteristic)
C1, C2, C2 [×2], 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, C3×C6 [×2], C2×Dic3 [×4], C2×C12 [×2], C4.Q8, C3×Dic3 [×2], C3⋊Dic3 [×2], C62, Dic3⋊C4 [×2], C322C8 [×2], C6×Dic3 [×2], C2×C3⋊Dic3, C62.C22 [×2], C2×C322C8, C62.6D4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4, Q8, C4⋊C4, SD16 [×2], C4.Q8, S3≀C2, C3⋊S3.Q8, C322SD16 [×2], C62.6D4

Character table of C62.6D4

 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 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 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 complex lifted from SD16 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 complex lifted from SD16 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 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⋊2SD16, 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⋊2SD16, 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⋊2SD16, 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 complex lifted from C32⋊2SD16 ρ24 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⋊2SD16 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⋊2SD16 ρ26 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 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 C62.6D4
On 96 points
Generators in S96
```(1 79)(2 16 26 80 52 88)(3 73)(4 82 54 74 28 10)(5 75)(6 12 30 76 56 84)(7 77)(8 86 50 78 32 14)(9 53)(11 55)(13 49)(15 51)(17 89 59 46 35 71)(18 47)(19 65 37 48 61 91)(20 41)(21 93 63 42 39 67)(22 43)(23 69 33 44 57 95)(24 45)(25 87)(27 81)(29 83)(31 85)(34 96)(36 90)(38 92)(40 94)(58 70)(60 72)(62 66)(64 68)
(1 55 25 5 51 29)(2 6)(3 31 53 7 27 49)(4 8)(9 77 81 13 73 85)(10 14)(11 87 75 15 83 79)(12 16)(17 21)(18 64 36 22 60 40)(19 23)(20 34 62 24 38 58)(26 30)(28 32)(33 37)(35 39)(41 96 66 45 92 70)(42 46)(43 72 94 47 68 90)(44 48)(50 54)(52 56)(57 61)(59 63)(65 69)(67 71)(74 78)(76 80)(82 86)(84 88)(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 47 79 18)(2 42 80 21)(3 45 73 24)(4 48 74 19)(5 43 75 22)(6 46 76 17)(7 41 77 20)(8 44 78 23)(9 58 53 70)(10 61 54 65)(11 64 55 68)(12 59 56 71)(13 62 49 66)(14 57 50 69)(15 60 51 72)(16 63 52 67)(25 90 87 36)(26 93 88 39)(27 96 81 34)(28 91 82 37)(29 94 83 40)(30 89 84 35)(31 92 85 38)(32 95 86 33)```

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

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

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

Matrix representation of C62.6D4 in GL6(𝔽73)

 72 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 45 18 1 1 0 0 0 0 72 0
,
 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 9 27 1 0 0 0 9 27 0 1
,
 19 60 0 0 0 0 34 42 0 0 0 0 0 0 26 25 46 19 0 0 26 25 19 46 0 0 47 68 72 23 0 0 20 68 72 23
,
 24 16 0 0 0 0 5 49 0 0 0 0 0 0 27 0 0 0 0 0 0 27 0 0 0 0 40 16 27 27 0 0 66 41 0 46

`G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,45,0,0,0,0,72,18,0,0,0,0,0,1,72,0,0,0,0,1,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,9,9,0,0,72,72,27,27,0,0,0,0,1,0,0,0,0,0,0,1],[19,34,0,0,0,0,60,42,0,0,0,0,0,0,26,26,47,20,0,0,25,25,68,68,0,0,46,19,72,72,0,0,19,46,23,23],[24,5,0,0,0,0,16,49,0,0,0,0,0,0,27,0,40,66,0,0,0,27,16,41,0,0,0,0,27,0,0,0,0,0,27,46] >;`

C62.6D4 in GAP, Magma, Sage, TeX

`C_6^2._6D_4`
`% in TeX`

`G:=Group("C6^2.6D4");`
`// GroupNames label`

`G:=SmallGroup(288,390);`
`// by ID`

`G=gap.SmallGroup(288,390);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,85,176,422,219,100,2693,2028,691,797,2372]);`
`// 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=c^3>;`
`// generators/relations`

Export

׿
×
𝔽