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## G = C6.1152+ 1+4order 192 = 26·3

### 24th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C4○D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C6.1152+ 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — C2×C3⋊D4 — C4×C3⋊D4 — C6.1152+ 1+4
 Lower central C3 — C2×C6 — C6.1152+ 1+4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C6.1152+ 1+4
G = < a,b,c,d,e | a6=b4=1, c2=e2=a3, d2=b2, ab=ba, cac-1=dad-1=eae-1=a-1, cbc-1=b-1, dbd-1=a3b, be=eb, cd=dc, ce=ec, ede-1=a3b2d >

Subgroups: 576 in 238 conjugacy classes, 99 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, C12, 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, C2×D4, C4×S3, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C2×C4⋊C4, C42⋊C2, C4×D4, C4⋊D4, C4⋊D4, C22.D4, C42.C2, C422C2, C4×Dic3, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, C22.47C24, C23.8D6, C23.21D6, C4.Dic6, C4⋊C47S3, C2×C4⋊Dic3, C4×C3⋊D4, D4×Dic3, D4×Dic3, D63D4, C23.14D6, C3×C4⋊D4, C6.1152+ 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, D4○D12, C6.1152+ 1+4

Smallest permutation representation of C6.1152+ 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 29 18 19)(2 30 13 20)(3 25 14 21)(4 26 15 22)(5 27 16 23)(6 28 17 24)(7 83 95 87)(8 84 96 88)(9 79 91 89)(10 80 92 90)(11 81 93 85)(12 82 94 86)(31 53 41 43)(32 54 42 44)(33 49 37 45)(34 50 38 46)(35 51 39 47)(36 52 40 48)(55 74 65 70)(56 75 66 71)(57 76 61 72)(58 77 62 67)(59 78 63 68)(60 73 64 69)
(1 79 4 82)(2 84 5 81)(3 83 6 80)(7 28 10 25)(8 27 11 30)(9 26 12 29)(13 88 16 85)(14 87 17 90)(15 86 18 89)(19 91 22 94)(20 96 23 93)(21 95 24 92)(31 55 34 58)(32 60 35 57)(33 59 36 56)(37 63 40 66)(38 62 41 65)(39 61 42 64)(43 74 46 77)(44 73 47 76)(45 78 48 75)(49 68 52 71)(50 67 53 70)(51 72 54 69)
(1 62 18 58)(2 61 13 57)(3 66 14 56)(4 65 15 55)(5 64 16 60)(6 63 17 59)(7 52 95 48)(8 51 96 47)(9 50 91 46)(10 49 92 45)(11 54 93 44)(12 53 94 43)(19 74 29 70)(20 73 30 69)(21 78 25 68)(22 77 26 67)(23 76 27 72)(24 75 28 71)(31 79 41 89)(32 84 42 88)(33 83 37 87)(34 82 38 86)(35 81 39 85)(36 80 40 90)
(1 43 4 46)(2 48 5 45)(3 47 6 44)(7 57 10 60)(8 56 11 59)(9 55 12 58)(13 52 16 49)(14 51 17 54)(15 50 18 53)(19 41 22 38)(20 40 23 37)(21 39 24 42)(25 35 28 32)(26 34 29 31)(27 33 30 36)(61 92 64 95)(62 91 65 94)(63 96 66 93)(67 89 70 86)(68 88 71 85)(69 87 72 90)(73 83 76 80)(74 82 77 79)(75 81 78 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), (1,29,18,19)(2,30,13,20)(3,25,14,21)(4,26,15,22)(5,27,16,23)(6,28,17,24)(7,83,95,87)(8,84,96,88)(9,79,91,89)(10,80,92,90)(11,81,93,85)(12,82,94,86)(31,53,41,43)(32,54,42,44)(33,49,37,45)(34,50,38,46)(35,51,39,47)(36,52,40,48)(55,74,65,70)(56,75,66,71)(57,76,61,72)(58,77,62,67)(59,78,63,68)(60,73,64,69), (1,79,4,82)(2,84,5,81)(3,83,6,80)(7,28,10,25)(8,27,11,30)(9,26,12,29)(13,88,16,85)(14,87,17,90)(15,86,18,89)(19,91,22,94)(20,96,23,93)(21,95,24,92)(31,55,34,58)(32,60,35,57)(33,59,36,56)(37,63,40,66)(38,62,41,65)(39,61,42,64)(43,74,46,77)(44,73,47,76)(45,78,48,75)(49,68,52,71)(50,67,53,70)(51,72,54,69), (1,62,18,58)(2,61,13,57)(3,66,14,56)(4,65,15,55)(5,64,16,60)(6,63,17,59)(7,52,95,48)(8,51,96,47)(9,50,91,46)(10,49,92,45)(11,54,93,44)(12,53,94,43)(19,74,29,70)(20,73,30,69)(21,78,25,68)(22,77,26,67)(23,76,27,72)(24,75,28,71)(31,79,41,89)(32,84,42,88)(33,83,37,87)(34,82,38,86)(35,81,39,85)(36,80,40,90), (1,43,4,46)(2,48,5,45)(3,47,6,44)(7,57,10,60)(8,56,11,59)(9,55,12,58)(13,52,16,49)(14,51,17,54)(15,50,18,53)(19,41,22,38)(20,40,23,37)(21,39,24,42)(25,35,28,32)(26,34,29,31)(27,33,30,36)(61,92,64,95)(62,91,65,94)(63,96,66,93)(67,89,70,86)(68,88,71,85)(69,87,72,90)(73,83,76,80)(74,82,77,79)(75,81,78,84)>;`

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

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

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 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + - - + image C1 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⋊2S3 D4○D12 kernel C6.1152+ 1+4 C23.8D6 C23.21D6 C4.Dic6 C4⋊C4⋊7S3 C2×C4⋊Dic3 C4×C3⋊D4 D4×Dic3 D6⋊3D4 C23.14D6 C3×C4⋊D4 C4⋊D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C12 C2×C6 C6 C4 C22 C2 # reps 1 2 2 1 1 1 1 3 1 2 1 1 2 1 1 3 4 4 1 2 2 2

Matrix representation of C6.1152+ 1+4 in GL6(𝔽13)

 12 0 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 0 12 0 0 0 0 1 12
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 5 0 0 0 0 0 4 8 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 5 0 0 0 0 0 0 5 0 0 0 0 0 0 8 6 0 0 0 0 0 5 0 0 0 0 0 0 0 12 0 0 0 0 12 0
,
 12 0 0 0 0 0 0 1 0 0 0 0 0 0 8 6 0 0 0 0 0 5 0 0 0 0 0 0 0 12 0 0 0 0 12 0
,
 0 5 0 0 0 0 5 0 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 0 12 0 0 0 0 12 0

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

C6.1152+ 1+4 in GAP, Magma, Sage, TeX

`C_6._{115}2_+^{1+4}`
`% in TeX`

`G:=Group("C6.115ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1177);`
`// by ID`

`G=gap.SmallGroup(192,1177);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,100,675,570,185,136,6278]);`
`// Polycyclic`

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

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