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## G = C6.822- 1+4order 192 = 26·3

### 37th 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 — C6.822- 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — C2×C3⋊D4 — C2×C4○D12 — C6.822- 1+4
 Lower central C3 — C2×C6 — C6.822- 1+4
 Upper central C1 — C22 — C22.D4

Generators and relations for C6.822- 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=e2=b2, bab-1=cac=dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=a3b2d >

Subgroups: 736 in 292 conjugacy classes, 105 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C22.D4, C4⋊Q8, C2×C4○D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, D42S3, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, D46D4, Dic3.D4, Dic34D4, C23.9D6, Dic3⋊D4, C23.21D6, C12⋊Q8, S3×C4⋊C4, D6.D4, D6⋊Q8, C2×Dic3⋊C4, C23.14D6, C3×C22.D4, C2×C4○D12, C2×D42S3, C6.822- 1+4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C22×S3, C22×D4, C2×C4○D4, 2- 1+4, S3×D4, S3×C23, D46D4, C2×S3×D4, S3×C4○D4, Q8○D12, C6.822- 1+4

Smallest permutation representation of C6.822- 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 34 18 38)(2 33 13 37)(3 32 14 42)(4 31 15 41)(5 36 16 40)(6 35 17 39)(7 72 95 76)(8 71 96 75)(9 70 91 74)(10 69 92 73)(11 68 93 78)(12 67 94 77)(19 50 29 46)(20 49 30 45)(21 54 25 44)(22 53 26 43)(23 52 27 48)(24 51 28 47)(55 79 65 89)(56 84 66 88)(57 83 61 87)(58 82 62 86)(59 81 63 85)(60 80 64 90)
(1 82)(2 81)(3 80)(4 79)(5 84)(6 83)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 85)(14 90)(15 89)(16 88)(17 87)(18 86)(25 92)(26 91)(27 96)(28 95)(29 94)(30 93)(31 55)(32 60)(33 59)(34 58)(35 57)(36 56)(37 63)(38 62)(39 61)(40 66)(41 65)(42 64)(43 70)(44 69)(45 68)(46 67)(47 72)(48 71)(49 78)(50 77)(51 76)(52 75)(53 74)(54 73)
(1 9 18 91)(2 8 13 96)(3 7 14 95)(4 12 15 94)(5 11 16 93)(6 10 17 92)(19 89 29 79)(20 88 30 84)(21 87 25 83)(22 86 26 82)(23 85 27 81)(24 90 28 80)(31 67 41 77)(32 72 42 76)(33 71 37 75)(34 70 38 74)(35 69 39 73)(36 68 40 78)(43 62 53 58)(44 61 54 57)(45 66 49 56)(46 65 50 55)(47 64 51 60)(48 63 52 59)
(1 34 18 38)(2 35 13 39)(3 36 14 40)(4 31 15 41)(5 32 16 42)(6 33 17 37)(7 75 95 71)(8 76 96 72)(9 77 91 67)(10 78 92 68)(11 73 93 69)(12 74 94 70)(19 53 29 43)(20 54 30 44)(21 49 25 45)(22 50 26 46)(23 51 27 47)(24 52 28 48)(55 89 65 79)(56 90 66 80)(57 85 61 81)(58 86 62 82)(59 87 63 83)(60 88 64 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), (1,34,18,38)(2,33,13,37)(3,32,14,42)(4,31,15,41)(5,36,16,40)(6,35,17,39)(7,72,95,76)(8,71,96,75)(9,70,91,74)(10,69,92,73)(11,68,93,78)(12,67,94,77)(19,50,29,46)(20,49,30,45)(21,54,25,44)(22,53,26,43)(23,52,27,48)(24,51,28,47)(55,79,65,89)(56,84,66,88)(57,83,61,87)(58,82,62,86)(59,81,63,85)(60,80,64,90), (1,82)(2,81)(3,80)(4,79)(5,84)(6,83)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,85)(14,90)(15,89)(16,88)(17,87)(18,86)(25,92)(26,91)(27,96)(28,95)(29,94)(30,93)(31,55)(32,60)(33,59)(34,58)(35,57)(36,56)(37,63)(38,62)(39,61)(40,66)(41,65)(42,64)(43,70)(44,69)(45,68)(46,67)(47,72)(48,71)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73), (1,9,18,91)(2,8,13,96)(3,7,14,95)(4,12,15,94)(5,11,16,93)(6,10,17,92)(19,89,29,79)(20,88,30,84)(21,87,25,83)(22,86,26,82)(23,85,27,81)(24,90,28,80)(31,67,41,77)(32,72,42,76)(33,71,37,75)(34,70,38,74)(35,69,39,73)(36,68,40,78)(43,62,53,58)(44,61,54,57)(45,66,49,56)(46,65,50,55)(47,64,51,60)(48,63,52,59), (1,34,18,38)(2,35,13,39)(3,36,14,40)(4,31,15,41)(5,32,16,42)(6,33,17,37)(7,75,95,71)(8,76,96,72)(9,77,91,67)(10,78,92,68)(11,73,93,69)(12,74,94,70)(19,53,29,43)(20,54,30,44)(21,49,25,45)(22,50,26,46)(23,51,27,47)(24,52,28,48)(55,89,65,79)(56,90,66,80)(57,85,61,81)(58,86,62,82)(59,87,63,83)(60,88,64,84)>;`

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G ··· 4L 4M 4N 4O 6A 6B 6C 6D 6E 6F 12A 12B 12C 12D 12E 12F 12G order 1 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 ··· 4 4 4 4 6 6 6 6 6 6 12 12 12 12 12 12 12 size 1 1 1 1 2 2 4 6 6 12 2 2 2 4 4 4 4 6 ··· 6 12 12 12 2 2 2 4 4 8 4 4 4 4 8 8 8

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 D6 C4○D4 2- 1+4 S3×D4 S3×C4○D4 Q8○D12 kernel C6.822- 1+4 Dic3.D4 Dic3⋊4D4 C23.9D6 Dic3⋊D4 C23.21D6 C12⋊Q8 S3×C4⋊C4 D6.D4 D6⋊Q8 C2×Dic3⋊C4 C23.14D6 C3×C22.D4 C2×C4○D12 C2×D4⋊2S3 C22.D4 C3⋊D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 Dic3 C6 C22 C2 C2 # reps 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 4 3 2 1 1 4 1 2 2 2

Matrix representation of C6.822- 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 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 5 0 0 0 0 0 10 8
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 1 12 0 0 0 0 0 12
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 8 0 0 0 0 0 0 8
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 5 0 0 0 0 0 0 5

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

C6.822- 1+4 in GAP, Magma, Sage, TeX

`C_6._{82}2_-^{1+4}`
`% in TeX`

`G:=Group("C6.82ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1214);`
`// by ID`

`G=gap.SmallGroup(192,1214);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,184,570,185,136,6278]);`
`// Polycyclic`

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

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