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## G = C6.82+ 1+4order 192 = 26·3

### 8th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C6.82+ 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C2×C4○D12 — C6.82+ 1+4
 Lower central C3 — C6 — C6.82+ 1+4
 Upper central C1 — C22 — C2×C4⋊C4

Generators and relations for C6.82+ 1+4
G = < a,b,c,d,e | a6=b4=1, c2=e2=a3, d2=a3b2, bab-1=dad-1=eae-1=a-1, ac=ca, cbc-1=a3b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=a3b2d >

Subgroups: 632 in 294 conjugacy classes, 151 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C22×S3, C22×C6, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C2×C4○D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, C22×C12, C23.33C23, Dic6⋊C4, S3×C4⋊C4, C4⋊C47S3, Dic35D4, C23.26D6, C4×C3⋊D4, C6×C4⋊C4, C2×C4○D12, C6.82+ 1+4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C24, C4×S3, C22×S3, C23×C4, 2+ 1+4, 2- 1+4, S3×C2×C4, S3×C23, C23.33C23, S3×C22×C4, D46D6, Q8.15D6, C6.82+ 1+4

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

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

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

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

54 conjugacy classes

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

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C4 S3 D6 D6 C4×S3 2+ 1+4 2- 1+4 D4⋊6D6 Q8.15D6 kernel C6.82+ 1+4 Dic6⋊C4 S3×C4⋊C4 C4⋊C4⋊7S3 Dic3⋊5D4 C23.26D6 C4×C3⋊D4 C6×C4⋊C4 C2×C4○D12 C4○D12 C2×C4⋊C4 C4⋊C4 C22×C4 C2×C4 C6 C6 C2 C2 # reps 1 2 2 2 2 1 4 1 1 16 1 4 3 8 1 1 2 2

Matrix representation of C6.82+ 1+4 in GL6(𝔽13)

 0 1 0 0 0 0 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 0 8 0 0 0 0 8 0 0 0 0 0 0 0 1 11 0 0 0 0 0 12 0 0 0 0 5 1 0 12 0 0 5 2 12 0
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 1 11 0 0 0 0 1 12 0 0 0 0 7 1 0 1 0 0 5 2 12 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 3 0 0 3 0 0 12 0 8 8 0 0 1 8 0 5 0 0 1 0 0 10
,
 0 8 0 0 0 0 8 0 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 5 0 5 0 0 0 10 0 0 5

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

C6.82+ 1+4 in GAP, Magma, Sage, TeX

`C_6._82_+^{1+4}`
`% in TeX`

`G:=Group("C6.8ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1063);`
`// by ID`

`G=gap.SmallGroup(192,1063);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,570,80,6278]);`
`// Polycyclic`

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