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## G = C6.1042- 1+4order 192 = 26·3

### 59th 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 — C6.1042- 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C2×Dic3⋊C4 — C6.1042- 1+4
 Lower central C3 — C2×C6 — C6.1042- 1+4
 Upper central C1 — C22 — C2×C4○D4

Generators and relations for C6.1042- 1+4
G = < a,b,c,d,e | a6=b4=1, c2=a3, d2=e2=b2, bab-1=cac-1=eae-1=a-1, ad=da, cbc-1=a3b-1, bd=db, be=eb, cd=dc, ece-1=a3c, ede-1=b2d >

Subgroups: 664 in 292 conjugacy classes, 113 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C42, 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, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×C6, C22×C6, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C4⋊Q8, C2×C4○D4, C2×C4○D4, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C6.D4, C2×Dic6, S3×C2×C4, D42S3, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, D46D4, C2×Dic3⋊C4, C12.48D4, C4×C3⋊D4, C23.28D6, D4×Dic3, C23.23D6, C23.14D6, Dic3⋊Q8, D63Q8, C2×D42S3, C6×C4○D4, C6.1042- 1+4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C3⋊D4, C22×S3, C22×D4, C2×C4○D4, 2- 1+4, C2×C3⋊D4, S3×C23, D46D4, S3×C4○D4, Q8○D12, C22×C3⋊D4, C6.1042- 1+4

Smallest permutation representation of C6.1042- 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 77 15 70)(2 76 16 69)(3 75 17 68)(4 74 18 67)(5 73 13 72)(6 78 14 71)(7 37 92 36)(8 42 93 35)(9 41 94 34)(10 40 95 33)(11 39 96 32)(12 38 91 31)(19 65 26 58)(20 64 27 57)(21 63 28 56)(22 62 29 55)(23 61 30 60)(24 66 25 59)(43 79 50 86)(44 84 51 85)(45 83 52 90)(46 82 53 89)(47 81 54 88)(48 80 49 87)
(1 38 4 41)(2 37 5 40)(3 42 6 39)(7 76 10 73)(8 75 11 78)(9 74 12 77)(13 33 16 36)(14 32 17 35)(15 31 18 34)(19 50 22 53)(20 49 23 52)(21 54 24 51)(25 44 28 47)(26 43 29 46)(27 48 30 45)(55 86 58 89)(56 85 59 88)(57 90 60 87)(61 80 64 83)(62 79 65 82)(63 84 66 81)(67 91 70 94)(68 96 71 93)(69 95 72 92)
(1 77 15 70)(2 78 16 71)(3 73 17 72)(4 74 18 67)(5 75 13 68)(6 76 14 69)(7 35 92 42)(8 36 93 37)(9 31 94 38)(10 32 95 39)(11 33 96 40)(12 34 91 41)(19 58 26 65)(20 59 27 66)(21 60 28 61)(22 55 29 62)(23 56 30 63)(24 57 25 64)(43 82 50 89)(44 83 51 90)(45 84 52 85)(46 79 53 86)(47 80 54 87)(48 81 49 88)
(1 58 15 65)(2 57 16 64)(3 56 17 63)(4 55 18 62)(5 60 13 61)(6 59 14 66)(7 49 92 48)(8 54 93 47)(9 53 94 46)(10 52 95 45)(11 51 96 44)(12 50 91 43)(19 70 26 77)(20 69 27 76)(21 68 28 75)(22 67 29 74)(23 72 30 73)(24 71 25 78)(31 79 38 86)(32 84 39 85)(33 83 40 90)(34 82 41 89)(35 81 42 88)(36 80 37 87)```

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

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

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

45 conjugacy classes

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

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 C4○D4 C3⋊D4 2- 1+4 S3×C4○D4 Q8○D12 kernel C6.1042- 1+4 C2×Dic3⋊C4 C12.48D4 C4×C3⋊D4 C23.28D6 D4×Dic3 C23.23D6 C23.14D6 Dic3⋊Q8 D6⋊3Q8 C2×D4⋊2S3 C6×C4○D4 C2×C4○D4 C3×D4 C22×C4 C2×D4 C2×Q8 Dic3 D4 C6 C2 C2 # reps 1 2 1 1 2 1 2 2 1 1 1 1 1 4 3 3 1 4 8 1 2 2

Matrix representation of C6.1042- 1+4 in GL6(𝔽13)

 12 12 0 0 0 0 1 0 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
,
 1 0 0 0 0 0 12 12 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 12 0 0 0 0 0 1 1
,
 1 0 0 0 0 0 12 12 0 0 0 0 0 0 5 0 0 0 0 0 5 8 0 0 0 0 0 0 1 2 0 0 0 0 12 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 5 0 0 0 0 0 5 8 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 1 1 0 0 0 0 0 0 5 3 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 12 12

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

C6.1042- 1+4 in GAP, Magma, Sage, TeX

`C_6._{104}2_-^{1+4}`
`% in TeX`

`G:=Group("C6.104ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1383);`
`// by ID`

`G=gap.SmallGroup(192,1383);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,758,100,346,6278]);`
`// Polycyclic`

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

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