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## G = C6.1082- 1+4order 192 = 26·3

### 63rd non-split extension by C6 of 2- 1+4 acting via 2- 1+4/C4○D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C6.1082- 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C2×C4○D12 — C6.1082- 1+4
 Lower central C3 — C2×C6 — C6.1082- 1+4
 Upper central C1 — C22 — C2×C4○D4

Generators and relations for C6.1082- 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=b2, e2=a3b2, bab-1=dad-1=a-1, ac=ca, ae=ea, cbc=b-1, bd=db, ebe-1=a3b, dcd-1=a3c, ce=ec, ede-1=b2d >

Subgroups: 744 in 294 conjugacy classes, 111 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, C22×C6, C2×C4⋊C4, C4⋊D4, C22⋊Q8, C2×C4○D4, C2×C4○D4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C22×Dic3, C2×C3⋊D4, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, C22.31C24, C12.48D4, C2×C4⋊Dic3, C127D4, D63D4, C23.14D6, D63Q8, C2×C4○D12, C6×C4○D4, C6.1082- 1+4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C3⋊D4, C22×S3, C22×D4, 2+ 1+4, 2- 1+4, C2×C3⋊D4, S3×C23, C22.31C24, D4○D12, Q8○D12, C22×C3⋊D4, C6.1082- 1+4

Smallest permutation representation of C6.1082- 1+4
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 22 15 29)(2 21 16 28)(3 20 17 27)(4 19 18 26)(5 24 13 25)(6 23 14 30)(7 84 92 85)(8 83 93 90)(9 82 94 89)(10 81 95 88)(11 80 96 87)(12 79 91 86)(31 53 38 46)(32 52 39 45)(33 51 40 44)(34 50 41 43)(35 49 42 48)(36 54 37 47)(55 67 62 74)(56 72 63 73)(57 71 64 78)(58 70 65 77)(59 69 66 76)(60 68 61 75)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 25)(7 87)(8 88)(9 89)(10 90)(11 85)(12 86)(13 23)(14 24)(15 19)(16 20)(17 21)(18 22)(31 53)(32 54)(33 49)(34 50)(35 51)(36 52)(37 45)(38 46)(39 47)(40 48)(41 43)(42 44)(55 77)(56 78)(57 73)(58 74)(59 75)(60 76)(61 69)(62 70)(63 71)(64 72)(65 67)(66 68)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)
(1 46 15 53)(2 45 16 52)(3 44 17 51)(4 43 18 50)(5 48 13 49)(6 47 14 54)(7 60 92 61)(8 59 93 66)(9 58 94 65)(10 57 95 64)(11 56 96 63)(12 55 91 62)(19 34 26 41)(20 33 27 40)(21 32 28 39)(22 31 29 38)(23 36 30 37)(24 35 25 42)(67 86 74 79)(68 85 75 84)(69 90 76 83)(70 89 77 82)(71 88 78 81)(72 87 73 80)
(1 94 18 12)(2 95 13 7)(3 96 14 8)(4 91 15 9)(5 92 16 10)(6 93 17 11)(19 89 29 79)(20 90 30 80)(21 85 25 81)(22 86 26 82)(23 87 27 83)(24 88 28 84)(31 67 41 77)(32 68 42 78)(33 69 37 73)(34 70 38 74)(35 71 39 75)(36 72 40 76)(43 55 53 65)(44 56 54 66)(45 57 49 61)(46 58 50 62)(47 59 51 63)(48 60 52 64)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G ··· 4L 6A 6B 6C 6D ··· 6I 12A 12B 12C 12D 12E ··· 12J order 1 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 ··· 4 6 6 6 6 ··· 6 12 12 12 12 12 ··· 12 size 1 1 1 1 2 2 4 4 12 12 2 2 2 2 2 4 4 12 ··· 12 2 2 2 4 ··· 4 2 2 2 2 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 C3⋊D4 2+ 1+4 2- 1+4 D4○D12 Q8○D12 kernel C6.1082- 1+4 C12.48D4 C2×C4⋊Dic3 C12⋊7D4 D6⋊3D4 C23.14D6 D6⋊3Q8 C2×C4○D12 C6×C4○D4 C2×C4○D4 C2×C12 C22×C4 C2×D4 C2×Q8 C2×C4 C6 C6 C2 C2 # reps 1 2 1 2 2 4 2 1 1 1 4 3 3 1 8 1 1 2 2

Matrix representation of C6.1082- 1+4 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 12 0 0 0 0 1 0 0 0 0 0 0 0 12 12 0 0 0 0 1 0
,
 0 12 0 0 0 0 1 0 0 0 0 0 0 0 11 9 0 0 0 0 11 2 0 0 0 0 2 0 11 9 0 0 11 11 11 2
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 11 9 0 0 0 0 4 2 0 0 0 0 3 10 2 4 0 0 3 6 9 11
,
 0 12 0 0 0 0 1 0 0 0 0 0 0 0 3 10 4 8 0 0 7 10 4 9 0 0 2 4 6 3 0 0 2 11 10 7
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 12 12 5 0 0 0 1 0 0 5

`G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,11,11,2,11,0,0,9,2,0,11,0,0,0,0,11,11,0,0,0,0,9,2],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,11,4,3,3,0,0,9,2,10,6,0,0,0,0,2,9,0,0,0,0,4,11],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,3,7,2,2,0,0,10,10,4,11,0,0,4,4,6,10,0,0,8,9,3,7],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,8,0,12,1,0,0,0,8,12,0,0,0,0,0,5,0,0,0,0,0,0,5] >;`

C6.1082- 1+4 in GAP, Magma, Sage, TeX

`C_6._{108}2_-^{1+4}`
`% in TeX`

`G:=Group("C6.108ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1392);`
`// by ID`

`G=gap.SmallGroup(192,1392);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,675,570,297,6278]);`
`// Polycyclic`

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

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