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## G = C6.2- 1+4order 192 = 26·3

### 3rd 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 — C2×C6 — C6.2- 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — S3×C4⋊C4 — C6.2- 1+4
 Lower central C3 — C2×C6 — C6.2- 1+4
 Upper central C1 — C22 — C2×C4⋊C4

Generators and relations for C6.2- 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=a3b-1, dbd-1=a3b, be=eb, dcd-1=a3c, ce=ec, ede-1=b2d >

Subgroups: 744 in 292 conjugacy classes, 107 normal (43 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, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C4⋊Q8, C2×C4○D4, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, C4○D12, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C22×C12, D46D4, C12⋊Q8, S3×C4⋊C4, Dic35D4, D6.D4, C12⋊D4, D6⋊Q8, C4×C3⋊D4, C23.28D6, C127D4, C6×C4⋊C4, C2×C4○D12, C6.2- 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, C4○D12, S3×D4, S3×C23, D46D4, C2×C4○D12, C2×S3×D4, Q8.15D6, C6.2- 1+4

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A ··· 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 6A ··· 6G 12A ··· 12L 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 12 ··· 12 size 1 1 1 1 2 2 6 6 12 12 2 2 ··· 2 4 4 4 6 6 12 12 12 12 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 4 4 4 type + + + + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 C4○D4 C4○D12 2- 1+4 S3×D4 Q8.15D6 kernel C6.2- 1+4 C12⋊Q8 S3×C4⋊C4 Dic3⋊5D4 D6.D4 C12⋊D4 D6⋊Q8 C4×C3⋊D4 C23.28D6 C12⋊7D4 C6×C4⋊C4 C2×C4○D12 C2×C4⋊C4 C3⋊D4 C4⋊C4 C22×C4 C12 C4 C6 C22 C2 # reps 1 1 1 1 2 1 2 1 2 1 1 2 1 4 4 3 4 8 1 2 2

Matrix representation of C6.2- 1+4 in GL4(𝔽13) generated by

 1 1 0 0 12 0 0 0 0 0 12 0 0 0 0 12
,
 11 9 0 0 4 2 0 0 0 0 6 2 0 0 1 7
,
 5 0 0 0 8 8 0 0 0 0 6 2 0 0 1 7
,
 11 9 0 0 11 2 0 0 0 0 12 0 0 0 6 1
,
 1 0 0 0 0 1 0 0 0 0 7 11 0 0 12 6
`G:=sub<GL(4,GF(13))| [1,12,0,0,1,0,0,0,0,0,12,0,0,0,0,12],[11,4,0,0,9,2,0,0,0,0,6,1,0,0,2,7],[5,8,0,0,0,8,0,0,0,0,6,1,0,0,2,7],[11,11,0,0,9,2,0,0,0,0,12,6,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,7,12,0,0,11,6] >;`

C6.2- 1+4 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(192,1066);`
`// by ID`

`G=gap.SmallGroup(192,1066);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,100,675,6278]);`
`// 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=a^3*b^-1,d*b*d^-1=a^3*b,b*e=e*b,d*c*d^-1=a^3*c,c*e=e*c,e*d*e^-1=b^2*d>;`
`// generators/relations`

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