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## G = C6.342+ 1+4order 192 = 26·3

### 34th 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 — C2×C6 — C6.342+ 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C2×Dic3⋊C4 — C6.342+ 1+4
 Lower central C3 — C2×C6 — C6.342+ 1+4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C6.342+ 1+4
G = < a,b,c,d,e | a6=b4=e2=1, c2=a3, d2=b2, bab-1=cac-1=a-1, ad=da, ae=ea, cbc-1=b-1, dbd-1=ebe=a3b, cd=dc, ce=ec, ede=a3b2d >

Subgroups: 576 in 238 conjugacy classes, 97 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C2×C4⋊C4, C42⋊C2, C4×D4, C4⋊D4, C4⋊D4, C22.D4, C42.C2, C422C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, C22.47C24, C23.16D6, C23.8D6, Dic34D4, C23.9D6, Dic3.Q8, C4⋊C4⋊S3, C2×Dic3⋊C4, C4×C3⋊D4, D4×Dic3, C23.23D6, C23.14D6, C3×C4⋊D4, C6.342+ 1+4
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, D42S3, S3×C23, C22.47C24, C2×D42S3, D46D6, S3×C4○D4, C6.342+ 1+4

Smallest permutation representation of C6.342+ 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 65 14 55)(2 64 15 60)(3 63 16 59)(4 62 17 58)(5 61 18 57)(6 66 13 56)(7 49 95 45)(8 54 96 44)(9 53 91 43)(10 52 92 48)(11 51 93 47)(12 50 94 46)(19 77 29 67)(20 76 30 72)(21 75 25 71)(22 74 26 70)(23 73 27 69)(24 78 28 68)(31 86 41 82)(32 85 42 81)(33 90 37 80)(34 89 38 79)(35 88 39 84)(36 87 40 83)
(1 38 4 41)(2 37 5 40)(3 42 6 39)(7 73 10 76)(8 78 11 75)(9 77 12 74)(13 35 16 32)(14 34 17 31)(15 33 18 36)(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 79 58 82)(56 84 59 81)(57 83 60 80)(61 87 64 90)(62 86 65 89)(63 85 66 88)(67 94 70 91)(68 93 71 96)(69 92 72 95)
(1 70 14 74)(2 71 15 75)(3 72 16 76)(4 67 17 77)(5 68 18 78)(6 69 13 73)(7 42 95 32)(8 37 96 33)(9 38 91 34)(10 39 92 35)(11 40 93 36)(12 41 94 31)(19 55 29 65)(20 56 30 66)(21 57 25 61)(22 58 26 62)(23 59 27 63)(24 60 28 64)(43 86 53 82)(44 87 54 83)(45 88 49 84)(46 89 50 79)(47 90 51 80)(48 85 52 81)
(1 12)(2 7)(3 8)(4 9)(5 10)(6 11)(13 93)(14 94)(15 95)(16 96)(17 91)(18 92)(19 89)(20 90)(21 85)(22 86)(23 87)(24 88)(25 81)(26 82)(27 83)(28 84)(29 79)(30 80)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 73)(38 74)(39 75)(40 76)(41 77)(42 78)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)```

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 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 S3 D6 D6 D6 D6 C4○D4 C4○D4 2+ 1+4 D4⋊2S3 D4⋊6D6 S3×C4○D4 kernel C6.342+ 1+4 C23.16D6 C23.8D6 Dic3⋊4D4 C23.9D6 Dic3.Q8 C4⋊C4⋊S3 C2×Dic3⋊C4 C4×C3⋊D4 D4×Dic3 C23.23D6 C23.14D6 C3×C4⋊D4 C4⋊D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 Dic3 C2×C6 C6 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 1 3 1 1 2 1 1 3 4 4 1 2 2 2

Matrix representation of C6.342+ 1+4 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 12 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 0 5 0 0 0 0 5 0
,
 5 0 0 0 0 0 0 5 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 5 0 0 0 0 0 0 8
,
 1 11 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 8 0 0 0 0 0 0 5
,
 5 3 0 0 0 0 5 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12

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

C6.342+ 1+4 in GAP, Magma, Sage, TeX

`C_6._{34}2_+^{1+4}`
`% in TeX`

`G:=Group("C6.34ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1160);`
`// by ID`

`G=gap.SmallGroup(192,1160);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,758,100,794,297,6278]);`
`// Polycyclic`

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

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