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## G = C6.732- 1+4order 192 = 26·3

### 28th 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 — C6.732- 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — S3×C4⋊C4 — C6.732- 1+4
 Lower central C3 — C2×C6 — C6.732- 1+4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C6.732- 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=b2, e2=a3b2, bab-1=dad-1=a-1, ac=ca, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=a3b2d >

Subgroups: 704 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, C23, Dic3, Dic3, C12, C12, D6, 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, C2×Q8, C4○D4, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C2×C4⋊C4, C4×D4, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4⋊Q8, C2×C4○D4, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C6.D4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, S3×C2×C4, D42S3, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, D46D4, Dic3.D4, C23.9D6, C12⋊Q8, S3×C4⋊C4, C2×C4⋊Dic3, C4×C3⋊D4, D4×Dic3, C23.23D6, D63D4, C3×C4⋊D4, C2×D42S3, C6.732- 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, S3×D4, D42S3, S3×C23, D46D4, C2×S3×D4, C2×D42S3, Q8○D12, C6.732- 1+4

Smallest permutation representation of C6.732- 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 22 15 29)(2 21 16 28)(3 20 17 27)(4 19 18 26)(5 24 13 25)(6 23 14 30)(7 81 92 88)(8 80 93 87)(9 79 94 86)(10 84 95 85)(11 83 96 90)(12 82 91 89)(31 46 38 53)(32 45 39 52)(33 44 40 51)(34 43 41 50)(35 48 42 49)(36 47 37 54)(55 77 62 70)(56 76 63 69)(57 75 64 68)(58 74 65 67)(59 73 66 72)(60 78 61 71)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 25)(7 80)(8 81)(9 82)(10 83)(11 84)(12 79)(13 23)(14 24)(15 19)(16 20)(17 21)(18 22)(31 50)(32 51)(33 52)(34 53)(35 54)(36 49)(37 48)(38 43)(39 44)(40 45)(41 46)(42 47)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(61 73)(62 74)(63 75)(64 76)(65 77)(66 78)(85 96)(86 91)(87 92)(88 93)(89 94)(90 95)
(1 31 15 38)(2 36 16 37)(3 35 17 42)(4 34 18 41)(5 33 13 40)(6 32 14 39)(7 76 92 69)(8 75 93 68)(9 74 94 67)(10 73 95 72)(11 78 96 71)(12 77 91 70)(19 43 26 50)(20 48 27 49)(21 47 28 54)(22 46 29 53)(23 45 30 52)(24 44 25 51)(55 82 62 89)(56 81 63 88)(57 80 64 87)(58 79 65 86)(59 84 66 85)(60 83 61 90)
(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), (1,22,15,29)(2,21,16,28)(3,20,17,27)(4,19,18,26)(5,24,13,25)(6,23,14,30)(7,81,92,88)(8,80,93,87)(9,79,94,86)(10,84,95,85)(11,83,96,90)(12,82,91,89)(31,46,38,53)(32,45,39,52)(33,44,40,51)(34,43,41,50)(35,48,42,49)(36,47,37,54)(55,77,62,70)(56,76,63,69)(57,75,64,68)(58,74,65,67)(59,73,66,72)(60,78,61,71), (1,26)(2,27)(3,28)(4,29)(5,30)(6,25)(7,80)(8,81)(9,82)(10,83)(11,84)(12,79)(13,23)(14,24)(15,19)(16,20)(17,21)(18,22)(31,50)(32,51)(33,52)(34,53)(35,54)(36,49)(37,48)(38,43)(39,44)(40,45)(41,46)(42,47)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(61,73)(62,74)(63,75)(64,76)(65,77)(66,78)(85,96)(86,91)(87,92)(88,93)(89,94)(90,95), (1,31,15,38)(2,36,16,37)(3,35,17,42)(4,34,18,41)(5,33,13,40)(6,32,14,39)(7,76,92,69)(8,75,93,68)(9,74,94,67)(10,73,95,72)(11,78,96,71)(12,77,91,70)(19,43,26,50)(20,48,27,49)(21,47,28,54)(22,46,29,53)(23,45,30,52)(24,44,25,51)(55,82,62,89)(56,81,63,88)(57,80,64,87)(58,79,65,86)(59,84,66,85)(60,83,61,90), (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), (1,22,15,29)(2,21,16,28)(3,20,17,27)(4,19,18,26)(5,24,13,25)(6,23,14,30)(7,81,92,88)(8,80,93,87)(9,79,94,86)(10,84,95,85)(11,83,96,90)(12,82,91,89)(31,46,38,53)(32,45,39,52)(33,44,40,51)(34,43,41,50)(35,48,42,49)(36,47,37,54)(55,77,62,70)(56,76,63,69)(57,75,64,68)(58,74,65,67)(59,73,66,72)(60,78,61,71), (1,26)(2,27)(3,28)(4,29)(5,30)(6,25)(7,80)(8,81)(9,82)(10,83)(11,84)(12,79)(13,23)(14,24)(15,19)(16,20)(17,21)(18,22)(31,50)(32,51)(33,52)(34,53)(35,54)(36,49)(37,48)(38,43)(39,44)(40,45)(41,46)(42,47)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(61,73)(62,74)(63,75)(64,76)(65,77)(66,78)(85,96)(86,91)(87,92)(88,93)(89,94)(90,95), (1,31,15,38)(2,36,16,37)(3,35,17,42)(4,34,18,41)(5,33,13,40)(6,32,14,39)(7,76,92,69)(8,75,93,68)(9,74,94,67)(10,73,95,72)(11,78,96,71)(12,77,91,70)(19,43,26,50)(20,48,27,49)(21,47,28,54)(22,46,29,53)(23,45,30,52)(24,44,25,51)(55,82,62,89)(56,81,63,88)(57,80,64,87)(58,79,65,86)(59,84,66,85)(60,83,61,90), (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)], [(1,22,15,29),(2,21,16,28),(3,20,17,27),(4,19,18,26),(5,24,13,25),(6,23,14,30),(7,81,92,88),(8,80,93,87),(9,79,94,86),(10,84,95,85),(11,83,96,90),(12,82,91,89),(31,46,38,53),(32,45,39,52),(33,44,40,51),(34,43,41,50),(35,48,42,49),(36,47,37,54),(55,77,62,70),(56,76,63,69),(57,75,64,68),(58,74,65,67),(59,73,66,72),(60,78,61,71)], [(1,26),(2,27),(3,28),(4,29),(5,30),(6,25),(7,80),(8,81),(9,82),(10,83),(11,84),(12,79),(13,23),(14,24),(15,19),(16,20),(17,21),(18,22),(31,50),(32,51),(33,52),(34,53),(35,54),(36,49),(37,48),(38,43),(39,44),(40,45),(41,46),(42,47),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72),(61,73),(62,74),(63,75),(64,76),(65,77),(66,78),(85,96),(86,91),(87,92),(88,93),(89,94),(90,95)], [(1,31,15,38),(2,36,16,37),(3,35,17,42),(4,34,18,41),(5,33,13,40),(6,32,14,39),(7,76,92,69),(8,75,93,68),(9,74,94,67),(10,73,95,72),(11,78,96,71),(12,77,91,70),(19,43,26,50),(20,48,27,49),(21,47,28,54),(22,46,29,53),(23,45,30,52),(24,44,25,51),(55,82,62,89),(56,81,63,88),(57,80,64,87),(58,79,65,86),(59,84,66,85),(60,83,61,90)], [(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)]])`

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F ··· 4K 4L 4M 4N 4O 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D 12E 12F 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 6 6 6 6 6 12 12 12 12 12 12 size 1 1 1 1 2 2 4 4 6 6 2 2 2 4 4 4 6 ··· 6 12 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 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 C2 S3 D4 D6 D6 D6 D6 C4○D4 2- 1+4 D4⋊2S3 S3×D4 Q8○D12 kernel C6.732- 1+4 Dic3.D4 C23.9D6 C12⋊Q8 S3×C4⋊C4 C2×C4⋊Dic3 C4×C3⋊D4 D4×Dic3 C23.23D6 D6⋊3D4 C3×C4⋊D4 C2×D4⋊2S3 C4⋊D4 C3⋊D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C12 C6 C4 C22 C2 # reps 1 2 2 1 1 1 1 1 2 1 1 2 1 4 2 1 1 3 4 1 2 2 2

Matrix representation of C6.732- 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
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 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
,
 10 6 0 0 0 0 7 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

`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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[10,7,0,0,0,0,6,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;`

C6.732- 1+4 in GAP, Magma, Sage, TeX

`C_6._{73}2_-^{1+4}`
`% in TeX`

`G:=Group("C6.73ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1170);`
`// by ID`

`G=gap.SmallGroup(192,1170);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,100,675,185,6278]);`
`// Polycyclic`

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