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## G = C6.152- 1+4order 192 = 26·3

### 15th 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 — C6.152- 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C4×Dic3 — Dic6⋊C4 — C6.152- 1+4
 Lower central C3 — C2×C6 — C6.152- 1+4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for C6.152- 1+4
G = < a,b,c,d,e | a6=b4=1, c2=a3, d2=e2=b2, ab=ba, cac-1=dad-1=a-1, ae=ea, cbc-1=b-1, dbd-1=a3b, be=eb, dcd-1=a3c, ce=ec, ede-1=b2d >

Subgroups: 384 in 192 conjugacy classes, 95 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C2×Q8, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×Q8, C22×C6, C42⋊C2, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C422C2, C4⋊Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C6.D4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C22×C12, C6×Q8, C22.35C24, C23.8D6, Dic6⋊C4, Dic3.Q8, C4.Dic6, C4.Dic6, C12.48D4, C23.26D6, Dic3⋊Q8, Q8×Dic3, C3×C22⋊Q8, C6.152- 1+4
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2- 1+4, D42S3, S3×C23, C22.35C24, C2×D42S3, Q8.15D6, Q8○D12, C6.152- 1+4

Smallest permutation representation of C6.152- 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 34 7 28)(2 35 8 29)(3 36 9 30)(4 31 10 25)(5 32 11 26)(6 33 12 27)(13 43 19 37)(14 44 20 38)(15 45 21 39)(16 46 22 40)(17 47 23 41)(18 48 24 42)(49 76 55 82)(50 77 56 83)(51 78 57 84)(52 73 58 79)(53 74 59 80)(54 75 60 81)(61 85 67 91)(62 86 68 92)(63 87 69 93)(64 88 70 94)(65 89 71 95)(66 90 72 96)
(1 80 4 83)(2 79 5 82)(3 84 6 81)(7 74 10 77)(8 73 11 76)(9 78 12 75)(13 95 16 92)(14 94 17 91)(15 93 18 96)(19 89 22 86)(20 88 23 85)(21 87 24 90)(25 50 28 53)(26 49 29 52)(27 54 30 51)(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 71 46 68)(44 70 47 67)(45 69 48 72)
(1 65 7 71)(2 64 8 70)(3 63 9 69)(4 62 10 68)(5 61 11 67)(6 66 12 72)(13 56 19 50)(14 55 20 49)(15 60 21 54)(16 59 22 53)(17 58 23 52)(18 57 24 51)(25 95 31 89)(26 94 32 88)(27 93 33 87)(28 92 34 86)(29 91 35 85)(30 96 36 90)(37 74 43 80)(38 73 44 79)(39 78 45 84)(40 77 46 83)(41 76 47 82)(42 75 48 81)
(1 31 7 25)(2 32 8 26)(3 33 9 27)(4 34 10 28)(5 35 11 29)(6 36 12 30)(13 43 19 37)(14 44 20 38)(15 45 21 39)(16 46 22 40)(17 47 23 41)(18 48 24 42)(49 79 55 73)(50 80 56 74)(51 81 57 75)(52 82 58 76)(53 83 59 77)(54 84 60 78)(61 91 67 85)(62 92 68 86)(63 93 69 87)(64 94 70 88)(65 95 71 89)(66 96 72 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), (1,34,7,28)(2,35,8,29)(3,36,9,30)(4,31,10,25)(5,32,11,26)(6,33,12,27)(13,43,19,37)(14,44,20,38)(15,45,21,39)(16,46,22,40)(17,47,23,41)(18,48,24,42)(49,76,55,82)(50,77,56,83)(51,78,57,84)(52,73,58,79)(53,74,59,80)(54,75,60,81)(61,85,67,91)(62,86,68,92)(63,87,69,93)(64,88,70,94)(65,89,71,95)(66,90,72,96), (1,80,4,83)(2,79,5,82)(3,84,6,81)(7,74,10,77)(8,73,11,76)(9,78,12,75)(13,95,16,92)(14,94,17,91)(15,93,18,96)(19,89,22,86)(20,88,23,85)(21,87,24,90)(25,50,28,53)(26,49,29,52)(27,54,30,51)(31,56,34,59)(32,55,35,58)(33,60,36,57)(37,65,40,62)(38,64,41,61)(39,63,42,66)(43,71,46,68)(44,70,47,67)(45,69,48,72), (1,65,7,71)(2,64,8,70)(3,63,9,69)(4,62,10,68)(5,61,11,67)(6,66,12,72)(13,56,19,50)(14,55,20,49)(15,60,21,54)(16,59,22,53)(17,58,23,52)(18,57,24,51)(25,95,31,89)(26,94,32,88)(27,93,33,87)(28,92,34,86)(29,91,35,85)(30,96,36,90)(37,74,43,80)(38,73,44,79)(39,78,45,84)(40,77,46,83)(41,76,47,82)(42,75,48,81), (1,31,7,25)(2,32,8,26)(3,33,9,27)(4,34,10,28)(5,35,11,29)(6,36,12,30)(13,43,19,37)(14,44,20,38)(15,45,21,39)(16,46,22,40)(17,47,23,41)(18,48,24,42)(49,79,55,73)(50,80,56,74)(51,81,57,75)(52,82,58,76)(53,83,59,77)(54,84,60,78)(61,91,67,85)(62,92,68,86)(63,93,69,87)(64,94,70,88)(65,95,71,89)(66,96,72,90)>;`

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C ··· 4G 4H 4I 4J 4K 4L ··· 4Q 6A 6B 6C 6D 6E 12A 12B 12C 12D 12E 12F 12G 12H order 1 2 2 2 2 3 4 4 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 2 2 2 4 ··· 4 6 6 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 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + - - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 D6 C4○D4 2- 1+4 D4⋊2S3 Q8.15D6 Q8○D12 kernel C6.152- 1+4 C23.8D6 Dic6⋊C4 Dic3.Q8 C4.Dic6 C12.48D4 C23.26D6 Dic3⋊Q8 Q8×Dic3 C3×C22⋊Q8 C22⋊Q8 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C12 C6 C4 C2 C2 # reps 1 4 1 2 3 1 1 1 1 1 1 2 3 1 1 4 2 2 2 2

Matrix representation of C6.152- 1+4 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 12 1 0 0 0 0 12 0 1 1 0 0 0 1 12 0
,
 12 0 0 0 0 0 0 1 0 0 0 0 0 0 3 7 0 0 0 0 6 10 0 0 0 0 6 0 10 7 0 0 0 7 6 3
,
 5 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 5 8 0 0 5 5 8 3 0 0 5 0 0 8 0 0 0 0 0 8
,
 0 5 0 0 0 0 8 0 0 0 0 0 0 0 2 9 0 0 0 0 11 11 0 0 0 0 2 0 9 11 0 0 11 9 2 4
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 10 6 0 0 0 0 7 3 0 0 0 0 0 6 10 7 0 0 7 0 6 3

`G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,12,0,0,0,1,1,0,1,0,0,0,0,1,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,3,6,6,0,0,0,7,10,0,7,0,0,0,0,10,6,0,0,0,0,7,3],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,0,5,5,0,0,0,0,5,0,0,0,0,5,8,0,0,0,0,8,3,8,8],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,2,11,2,11,0,0,9,11,0,9,0,0,0,0,9,2,0,0,0,0,11,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,7,0,7,0,0,6,3,6,0,0,0,0,0,10,6,0,0,0,0,7,3] >;`

C6.152- 1+4 in GAP, Magma, Sage, TeX

`C_6._{15}2_-^{1+4}`
`% in TeX`

`G:=Group("C6.15ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1184);`
`// by ID`

`G=gap.SmallGroup(192,1184);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,219,268,675,297,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^6=b^4=1,c^2=a^3,d^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,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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