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## G = C6.792- 1+4order 192 = 26·3

### 34th 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 — C6.792- 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C2×D4⋊2S3 — C6.792- 1+4
 Lower central C3 — C2×C6 — C6.792- 1+4
 Upper central C1 — C22 — C22.D4

Generators and relations for C6.792- 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=e2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, ebe-1=a3b, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 656 in 270 conjugacy classes, 103 normal (39 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C42⋊C2, C22⋊Q8, C22.D4, C22.D4, C4.4D4, C4⋊Q8, C22×Q8, C2×C4○D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C6.D4, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, C2×Dic6, S3×C2×C4, D42S3, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, C23.38C23, C23.16D6, Dic3.D4, C23.11D6, C23.21D6, C12⋊Q8, D6⋊Q8, C23.28D6, C23.23D6, C3×C22.D4, C22×Dic6, C2×D42S3, C6.792- 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○D12, C6.792- 1+4

Smallest permutation representation of C6.792- 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 89 17 84)(2 90 18 79)(3 85 13 80)(4 86 14 81)(5 87 15 82)(6 88 16 83)(7 27 94 22)(8 28 95 23)(9 29 96 24)(10 30 91 19)(11 25 92 20)(12 26 93 21)(31 66 42 55)(32 61 37 56)(33 62 38 57)(34 63 39 58)(35 64 40 59)(36 65 41 60)(43 78 54 67)(44 73 49 68)(45 74 50 69)(46 75 51 70)(47 76 52 71)(48 77 53 72)
(1 60)(2 55)(3 56)(4 57)(5 58)(6 59)(7 51)(8 52)(9 53)(10 54)(11 49)(12 50)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)
(1 84 17 89)(2 83 18 88)(3 82 13 87)(4 81 14 86)(5 80 15 85)(6 79 16 90)(7 28 94 23)(8 27 95 22)(9 26 96 21)(10 25 91 20)(11 30 92 19)(12 29 93 24)(31 64 42 59)(32 63 37 58)(33 62 38 57)(34 61 39 56)(35 66 40 55)(36 65 41 60)(43 68 54 73)(44 67 49 78)(45 72 50 77)(46 71 51 76)(47 70 52 75)(48 69 53 74)
(1 29 17 24)(2 30 18 19)(3 25 13 20)(4 26 14 21)(5 27 15 22)(6 28 16 23)(7 90 94 79)(8 85 95 80)(9 86 96 81)(10 87 91 82)(11 88 92 83)(12 89 93 84)(31 51 42 46)(32 52 37 47)(33 53 38 48)(34 54 39 43)(35 49 40 44)(36 50 41 45)(55 78 66 67)(56 73 61 68)(57 74 62 69)(58 75 63 70)(59 76 64 71)(60 77 65 72)```

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A ··· 4E 4F 4G 4H 4I 4J ··· 4N 6A 6B 6C 6D 6E 6F 12A 12B 12C 12D 12E 12F 12G order 1 2 2 2 2 2 2 2 3 4 ··· 4 4 4 4 4 4 ··· 4 6 6 6 6 6 6 12 12 12 12 12 12 12 size 1 1 1 1 2 2 4 12 2 4 ··· 4 6 6 6 6 12 ··· 12 2 2 2 4 4 8 4 4 4 4 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 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○D12 kernel C6.792- 1+4 C23.16D6 Dic3.D4 C23.11D6 C23.21D6 C12⋊Q8 D6⋊Q8 C23.28D6 C23.23D6 C3×C22.D4 C22×Dic6 C2×D4⋊2S3 C22.D4 C2×Dic3 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C6 C22 C2 # reps 1 1 2 2 1 2 2 1 1 1 1 1 1 4 3 2 1 1 2 2 4

Matrix representation of C6.792- 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 4 2 11 9 0 0 11 2 4 2 0 0 4 2 9 11 0 0 11 2 2 11
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 9 11 6 0 0 0 2 11 0 6 0 0 9 11 4 2 0 0 2 11 11 2
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 9 11 2 4 0 0 2 4 2 11 0 0 0 0 11 2 0 0 0 0 4 2
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 10 7 0 0 0 0 6 3 0 0 0 0 0 0 10 7 0 0 0 0 6 3

`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],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,4,11,4,11,0,0,2,2,2,2,0,0,11,4,9,2,0,0,9,2,11,11],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,9,2,9,2,0,0,11,11,11,11,0,0,6,0,4,11,0,0,0,6,2,2],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,9,2,0,0,0,0,11,4,0,0,0,0,2,2,11,4,0,0,4,11,2,2],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,10,6,0,0,0,0,7,3,0,0,0,0,0,0,10,6,0,0,0,0,7,3] >;`

C6.792- 1+4 in GAP, Magma, Sage, TeX

`C_6._{79}2_-^{1+4}`
`% in TeX`

`G:=Group("C6.79ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1207);`
`// by ID`

`G=gap.SmallGroup(192,1207);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,387,1123,185,136,6278]);`
`// 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,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=a^3*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;`
`// generators/relations`

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