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## G = C6.422- 1+4order 192 = 26·3

### 42nd non-split extension by C6 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C6.422- 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C4×Dic3 — Q8×Dic3 — C6.422- 1+4
 Lower central C3 — C6 — C6.422- 1+4
 Upper central C1 — C22 — C22×Q8

Generators and relations for C6.422- 1+4
G = < a,b,c,d,e | a6=b4=1, c2=a3, d2=a3b2, e2=b2, ab=ba, cac-1=dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 424 in 266 conjugacy classes, 191 normal (11 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, Q8, C23, Dic3, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C2×Dic3, C2×C12, C3×Q8, C22×C6, C42⋊C2, C4×Q8, C22×Q8, C4×Dic3, C4⋊Dic3, C6.D4, C22×C12, C6×Q8, C23.32C23, C23.26D6, Q8×Dic3, Q8×C2×C6, C6.422- 1+4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C24, C2×Dic3, C22×S3, C23×C4, 2- 1+4, C22×Dic3, S3×C23, C23.32C23, Q8.15D6, C23×Dic3, C6.422- 1+4

Smallest permutation representation of C6.422- 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 38 17 31)(2 39 18 32)(3 40 13 33)(4 41 14 34)(5 42 15 35)(6 37 16 36)(7 71 92 73)(8 72 93 74)(9 67 94 75)(10 68 95 76)(11 69 96 77)(12 70 91 78)(19 54 26 46)(20 49 27 47)(21 50 28 48)(22 51 29 43)(23 52 30 44)(24 53 25 45)(55 82 63 90)(56 83 64 85)(57 84 65 86)(58 79 66 87)(59 80 61 88)(60 81 62 89)
(1 85 4 88)(2 90 5 87)(3 89 6 86)(7 22 10 19)(8 21 11 24)(9 20 12 23)(13 81 16 84)(14 80 17 83)(15 79 18 82)(25 93 28 96)(26 92 29 95)(27 91 30 94)(31 56 34 59)(32 55 35 58)(33 60 36 57)(37 65 40 62)(38 64 41 61)(39 63 42 66)(43 68 46 71)(44 67 47 70)(45 72 48 69)(49 78 52 75)(50 77 53 74)(51 76 54 73)
(1 71 14 76)(2 70 15 75)(3 69 16 74)(4 68 17 73)(5 67 18 78)(6 72 13 77)(7 41 95 31)(8 40 96 36)(9 39 91 35)(10 38 92 34)(11 37 93 33)(12 42 94 32)(19 64 29 59)(20 63 30 58)(21 62 25 57)(22 61 26 56)(23 66 27 55)(24 65 28 60)(43 80 54 85)(44 79 49 90)(45 84 50 89)(46 83 51 88)(47 82 52 87)(48 81 53 86)
(1 41 17 34)(2 42 18 35)(3 37 13 36)(4 38 14 31)(5 39 15 32)(6 40 16 33)(7 76 92 68)(8 77 93 69)(9 78 94 70)(10 73 95 71)(11 74 96 72)(12 75 91 67)(19 51 26 43)(20 52 27 44)(21 53 28 45)(22 54 29 46)(23 49 30 47)(24 50 25 48)(55 87 63 79)(56 88 64 80)(57 89 65 81)(58 90 66 82)(59 85 61 83)(60 86 62 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), (1,38,17,31)(2,39,18,32)(3,40,13,33)(4,41,14,34)(5,42,15,35)(6,37,16,36)(7,71,92,73)(8,72,93,74)(9,67,94,75)(10,68,95,76)(11,69,96,77)(12,70,91,78)(19,54,26,46)(20,49,27,47)(21,50,28,48)(22,51,29,43)(23,52,30,44)(24,53,25,45)(55,82,63,90)(56,83,64,85)(57,84,65,86)(58,79,66,87)(59,80,61,88)(60,81,62,89), (1,85,4,88)(2,90,5,87)(3,89,6,86)(7,22,10,19)(8,21,11,24)(9,20,12,23)(13,81,16,84)(14,80,17,83)(15,79,18,82)(25,93,28,96)(26,92,29,95)(27,91,30,94)(31,56,34,59)(32,55,35,58)(33,60,36,57)(37,65,40,62)(38,64,41,61)(39,63,42,66)(43,68,46,71)(44,67,47,70)(45,72,48,69)(49,78,52,75)(50,77,53,74)(51,76,54,73), (1,71,14,76)(2,70,15,75)(3,69,16,74)(4,68,17,73)(5,67,18,78)(6,72,13,77)(7,41,95,31)(8,40,96,36)(9,39,91,35)(10,38,92,34)(11,37,93,33)(12,42,94,32)(19,64,29,59)(20,63,30,58)(21,62,25,57)(22,61,26,56)(23,66,27,55)(24,65,28,60)(43,80,54,85)(44,79,49,90)(45,84,50,89)(46,83,51,88)(47,82,52,87)(48,81,53,86), (1,41,17,34)(2,42,18,35)(3,37,13,36)(4,38,14,31)(5,39,15,32)(6,40,16,33)(7,76,92,68)(8,77,93,69)(9,78,94,70)(10,73,95,71)(11,74,96,72)(12,75,91,67)(19,51,26,43)(20,52,27,44)(21,53,28,45)(22,54,29,46)(23,49,30,47)(24,50,25,48)(55,87,63,79)(56,88,64,80)(57,89,65,81)(58,90,66,82)(59,85,61,83)(60,86,62,84)>;`

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A ··· 4L 4M ··· 4AB 6A ··· 6G 12A ··· 12L order 1 2 2 2 2 2 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 2 2 ··· 2 6 ··· 6 2 ··· 2 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + - + - image C1 C2 C2 C2 C4 S3 D6 Dic3 D6 2- 1+4 Q8.15D6 kernel C6.422- 1+4 C23.26D6 Q8×Dic3 Q8×C2×C6 C6×Q8 C22×Q8 C22×C4 C2×Q8 C2×Q8 C6 C2 # reps 1 6 8 1 16 1 3 8 4 2 4

Matrix representation of C6.422- 1+4 in GL6(𝔽13)

 1 1 0 0 0 0 12 0 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 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 5 0 0 0 0 5 0 0 0 0 0 1 1 5 2 0 0 8 8 0 8
,
 2 4 0 0 0 0 2 11 0 0 0 0 0 0 1 1 5 2 0 0 0 0 8 0 0 0 0 5 0 0 0 0 0 0 0 12
,
 11 9 0 0 0 0 11 2 0 0 0 0 0 0 8 8 1 3 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 5 12 5
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 5 0 0 0 0 5 0 0 0 0 0 12 12 8 11 0 0 0 0 0 5

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

C6.422- 1+4 in GAP, Magma, Sage, TeX

`C_6._{42}2_-^{1+4}`
`% in TeX`

`G:=Group("C6.42ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1371);`
`// by ID`

`G=gap.SmallGroup(192,1371);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,232,387,184,1123,6278]);`
`// Polycyclic`

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

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