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## G = C6.162- 1+4order 192 = 26·3

### 16th 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 — C6.162- 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C2×S3×Q8 — C6.162- 1+4
 Lower central C3 — C2×C6 — C6.162- 1+4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for C6.162- 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=e2=b2, ab=ba, cac=dad-1=a-1, ae=ea, cbc=b-1, dbd-1=a3b, be=eb, dcd-1=a3c, ce=ec, ede-1=b2d >

Subgroups: 672 in 270 conjugacy classes, 103 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, 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, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×Q8, C22×S3, C22×C6, C42⋊C2, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C22×Q8, C2×C4○D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, C4○D12, S3×Q8, C2×C3⋊D4, C22×C12, C6×Q8, C23.38C23, C23.9D6, C23.11D6, C12⋊Q8, C4⋊C47S3, D6.D4, C4.D12, C12.48D4, D63Q8, C3×C22⋊Q8, C2×C4○D12, C2×S3×Q8, C6.162- 1+4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, 2- 1+4, S3×D4, S3×C23, C23.38C23, C2×S3×D4, Q8.15D6, Q8○D12, C6.162- 1+4

Smallest permutation representation of C6.162- 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 7 61)(2 68 8 62)(3 69 9 63)(4 70 10 64)(5 71 11 65)(6 72 12 66)(13 55 19 49)(14 56 20 50)(15 57 21 51)(16 58 22 52)(17 59 23 53)(18 60 24 54)(25 94 31 88)(26 95 32 89)(27 96 33 90)(28 91 34 85)(29 92 35 86)(30 93 36 87)(37 82 43 76)(38 83 44 77)(39 84 45 78)(40 79 46 73)(41 80 47 74)(42 81 48 75)
(1 7)(2 12)(3 11)(4 10)(5 9)(6 8)(14 18)(15 17)(20 24)(21 23)(25 28)(26 27)(29 30)(31 34)(32 33)(35 36)(37 46)(38 45)(39 44)(40 43)(41 48)(42 47)(49 55)(50 60)(51 59)(52 58)(53 57)(54 56)(62 66)(63 65)(68 72)(69 71)(73 76)(74 75)(77 78)(79 82)(80 81)(83 84)(85 94)(86 93)(87 92)(88 91)(89 96)(90 95)
(1 91 7 85)(2 96 8 90)(3 95 9 89)(4 94 10 88)(5 93 11 87)(6 92 12 86)(13 73 19 79)(14 78 20 84)(15 77 21 83)(16 76 22 82)(17 75 23 81)(18 74 24 80)(25 67 31 61)(26 72 32 66)(27 71 33 65)(28 70 34 64)(29 69 35 63)(30 68 36 62)(37 49 43 55)(38 54 44 60)(39 53 45 59)(40 52 46 58)(41 51 47 57)(42 50 48 56)
(1 55 7 49)(2 56 8 50)(3 57 9 51)(4 58 10 52)(5 59 11 53)(6 60 12 54)(13 67 19 61)(14 68 20 62)(15 69 21 63)(16 70 22 64)(17 71 23 65)(18 72 24 66)(25 79 31 73)(26 80 32 74)(27 81 33 75)(28 82 34 76)(29 83 35 77)(30 84 36 78)(37 91 43 85)(38 92 44 86)(39 93 45 87)(40 94 46 88)(41 95 47 89)(42 96 48 90)```

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 D6 2- 1+4 S3×D4 Q8.15D6 Q8○D12 kernel C6.162- 1+4 C23.9D6 C23.11D6 C12⋊Q8 C4⋊C4⋊7S3 D6.D4 C4.D12 C12.48D4 D6⋊3Q8 C3×C22⋊Q8 C2×C4○D12 C2×S3×Q8 C22⋊Q8 C4×S3 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C6 C4 C2 C2 # reps 1 2 2 2 1 2 1 1 1 1 1 1 1 4 2 3 1 1 2 2 2 2

Matrix representation of C6.162- 1+4 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 12 0 0 0 0 1 0 0 0 0 0 0 0 12 12 0 0 0 0 1 0
,
 12 0 0 0 0 0 0 1 0 0 0 0 0 0 8 3 10 0 0 0 10 5 0 10 0 0 10 0 5 10 0 0 0 10 3 8
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 12 12
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 10 0 5 10 0 0 3 3 5 8 0 0 8 3 10 0 0 0 8 5 3 3
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 7 1 0 0 0 4 12 6 0 0 7 1 9 0 0 0 12 6 0 9

`G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,8,10,10,0,0,0,3,5,0,10,0,0,10,0,5,3,0,0,0,10,10,8],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,10,3,8,8,0,0,0,3,3,5,0,0,5,5,10,3,0,0,10,8,0,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,7,12,0,0,0,4,1,6,0,0,7,12,9,0,0,0,1,6,0,9] >;`

C6.162- 1+4 in GAP, Magma, Sage, TeX

`C_6._{16}2_-^{1+4}`
`% in TeX`

`G:=Group("C6.16ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1187);`
`// by ID`

`G=gap.SmallGroup(192,1187);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,100,675,297,6278]);`
`// Polycyclic`

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