Copied to
clipboard

## G = C62.6Q8order 288 = 25·32

### 3rd non-split extension by C62 of Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62.6Q8
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C2×C62 — Dic3×C2×C6 — C62.6Q8
 Lower central C32 — C3×C6 — C62.6Q8
 Upper central C1 — C23

Generators and relations for C62.6Q8
G = < a,b,c,d | a6=b6=c4=1, d2=a3b3c2, ab=ba, cac-1=a-1, ad=da, bc=cb, dbd-1=b-1, dcd-1=a3c-1 >

Subgroups: 530 in 179 conjugacy classes, 80 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C6, C6, C2×C4, C23, C32, Dic3, C12, C2×C6, C2×C6, C22×C4, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C22×C6, C22×C6, C2.C42, C3×Dic3, C3⋊Dic3, C62, C62, C22×Dic3, C22×Dic3, C22×C12, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C2×C3⋊Dic3, C2×C62, C6.C42, Dic3×C2×C6, C22×C3⋊Dic3, C62.6Q8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C42, C22⋊C4, C4⋊C4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2.C42, S32, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, S3×Dic3, C6.D6, D6⋊S3, C3⋊D12, C322Q8, C6.C42, Dic32, D6⋊Dic3, C6.D12, Dic3⋊Dic3, C62.C22, C62.6Q8

Smallest permutation representation of C62.6Q8
On 96 points
Generators in S96
```(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 12 5 10 3 8)(2 7 6 11 4 9)(13 90 17 88 15 86)(14 85 18 89 16 87)(19 26 23 30 21 28)(20 27 24 25 22 29)(31 43 33 45 35 47)(32 44 34 46 36 48)(37 52 39 54 41 50)(38 53 40 49 42 51)(55 72 57 68 59 70)(56 67 58 69 60 71)(61 75 63 77 65 73)(62 76 64 78 66 74)(79 95 83 93 81 91)(80 96 84 94 82 92)
(1 37 22 43)(2 42 23 48)(3 41 24 47)(4 40 19 46)(5 39 20 45)(6 38 21 44)(7 51 30 32)(8 50 25 31)(9 49 26 36)(10 54 27 35)(11 53 28 34)(12 52 29 33)(13 58 80 75)(14 57 81 74)(15 56 82 73)(16 55 83 78)(17 60 84 77)(18 59 79 76)(61 86 67 92)(62 85 68 91)(63 90 69 96)(64 89 70 95)(65 88 71 94)(66 87 72 93)
(1 78 30 71)(2 73 25 72)(3 74 26 67)(4 75 27 68)(5 76 28 69)(6 77 29 70)(7 65 22 55)(8 66 23 56)(9 61 24 57)(10 62 19 58)(11 63 20 59)(12 64 21 60)(13 51 91 43)(14 52 92 44)(15 53 93 45)(16 54 94 46)(17 49 95 47)(18 50 96 48)(31 90 42 79)(32 85 37 80)(33 86 38 81)(34 87 39 82)(35 88 40 83)(36 89 41 84)```

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 3A 3B 3C 4A ··· 4H 4I 4J 4K 4L 6A ··· 6N 6O ··· 6U 12A ··· 12P order 1 2 ··· 2 3 3 3 4 ··· 4 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 2 4 6 ··· 6 18 18 18 18 2 ··· 2 4 ··· 4 6 ··· 6

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + - - + - + + - + - + - image C1 C2 C2 C4 C4 S3 D4 Q8 Dic3 D6 Dic6 C4×S3 D12 C3⋊D4 S32 S3×Dic3 C6.D6 D6⋊S3 C3⋊D12 C32⋊2Q8 kernel C62.6Q8 Dic3×C2×C6 C22×C3⋊Dic3 C6×Dic3 C2×C3⋊Dic3 C22×Dic3 C62 C62 C2×Dic3 C22×C6 C2×C6 C2×C6 C2×C6 C2×C6 C23 C22 C22 C22 C22 C22 # reps 1 2 1 8 4 2 3 1 4 2 4 8 4 8 1 2 1 1 2 1

Matrix representation of C62.6Q8 in GL8(𝔽13)

 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 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 5 0 0 0 0 0 0 0 8 8 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 5 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 12
,
 12 11 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 5 5 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

`G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[5,8,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12],[12,1,0,0,0,0,0,0,11,1,0,0,0,0,0,0,0,0,8,5,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;`

C62.6Q8 in GAP, Magma, Sage, TeX

`C_6^2._6Q_8`
`% in TeX`

`G:=Group("C6^2.6Q8");`
`// GroupNames label`

`G:=SmallGroup(288,227);`
`// by ID`

`G=gap.SmallGroup(288,227);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,253,36,1356,9414]);`
`// Polycyclic`

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

׿
×
𝔽