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## G = C6.802- 1+4order 192 = 26·3

### 35th 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 — C6.802- 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C2×Dic3⋊C4 — C6.802- 1+4
 Lower central C3 — C2×C6 — C6.802- 1+4
 Upper central C1 — C22 — C22.D4

Generators and relations for C6.802- 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=e2=a3b2, bab-1=dad-1=a-1, ac=ca, ae=ea, cbc=a3b-1, bd=db, be=eb, cd=dc, ece-1=a3c, ede-1=b2d >

Subgroups: 448 in 214 conjugacy classes, 97 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, D4, Q8, C23, Dic3, Dic3, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×C6, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C22.D4, C42.C2, C422C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C22×Dic3, C22×C12, C6×D4, C22.46C24, C23.16D6, Dic3.D4, C23.8D6, Dic6⋊C4, Dic3.Q8, C4.Dic6, C2×Dic3⋊C4, C23.26D6, D4×Dic3, C23.23D6, C3×C22.D4, C6.802- 1+4
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2- 1+4, D42S3, S3×C23, C22.46C24, C2×D42S3, S3×C4○D4, Q8○D12, C6.802- 1+4

Smallest permutation representation of C6.802- 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 68 15 76)(2 67 16 75)(3 72 17 74)(4 71 18 73)(5 70 13 78)(6 69 14 77)(7 38 92 31)(8 37 93 36)(9 42 94 35)(10 41 95 34)(11 40 96 33)(12 39 91 32)(19 56 26 64)(20 55 27 63)(21 60 28 62)(22 59 29 61)(23 58 30 66)(24 57 25 65)(43 80 51 88)(44 79 52 87)(45 84 53 86)(46 83 54 85)(47 82 49 90)(48 81 50 89)
(7 10)(8 11)(9 12)(19 29)(20 30)(21 25)(22 26)(23 27)(24 28)(31 38)(32 39)(33 40)(34 41)(35 42)(36 37)(43 46)(44 47)(45 48)(49 52)(50 53)(51 54)(67 78)(68 73)(69 74)(70 75)(71 76)(72 77)(79 87)(80 88)(81 89)(82 90)(83 85)(84 86)(91 94)(92 95)(93 96)
(1 56 18 61)(2 55 13 66)(3 60 14 65)(4 59 15 64)(5 58 16 63)(6 57 17 62)(7 51 95 46)(8 50 96 45)(9 49 91 44)(10 54 92 43)(11 53 93 48)(12 52 94 47)(19 71 29 76)(20 70 30 75)(21 69 25 74)(22 68 26 73)(23 67 27 78)(24 72 28 77)(31 80 41 85)(32 79 42 90)(33 84 37 89)(34 83 38 88)(35 82 39 87)(36 81 40 86)
(1 54 18 43)(2 49 13 44)(3 50 14 45)(4 51 15 46)(5 52 16 47)(6 53 17 48)(7 61 95 56)(8 62 96 57)(9 63 91 58)(10 64 92 59)(11 65 93 60)(12 66 94 55)(19 31 29 41)(20 32 30 42)(21 33 25 37)(22 34 26 38)(23 35 27 39)(24 36 28 40)(67 90 78 79)(68 85 73 80)(69 86 74 81)(70 87 75 82)(71 88 76 83)(72 89 77 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), (1,68,15,76)(2,67,16,75)(3,72,17,74)(4,71,18,73)(5,70,13,78)(6,69,14,77)(7,38,92,31)(8,37,93,36)(9,42,94,35)(10,41,95,34)(11,40,96,33)(12,39,91,32)(19,56,26,64)(20,55,27,63)(21,60,28,62)(22,59,29,61)(23,58,30,66)(24,57,25,65)(43,80,51,88)(44,79,52,87)(45,84,53,86)(46,83,54,85)(47,82,49,90)(48,81,50,89), (7,10)(8,11)(9,12)(19,29)(20,30)(21,25)(22,26)(23,27)(24,28)(31,38)(32,39)(33,40)(34,41)(35,42)(36,37)(43,46)(44,47)(45,48)(49,52)(50,53)(51,54)(67,78)(68,73)(69,74)(70,75)(71,76)(72,77)(79,87)(80,88)(81,89)(82,90)(83,85)(84,86)(91,94)(92,95)(93,96), (1,56,18,61)(2,55,13,66)(3,60,14,65)(4,59,15,64)(5,58,16,63)(6,57,17,62)(7,51,95,46)(8,50,96,45)(9,49,91,44)(10,54,92,43)(11,53,93,48)(12,52,94,47)(19,71,29,76)(20,70,30,75)(21,69,25,74)(22,68,26,73)(23,67,27,78)(24,72,28,77)(31,80,41,85)(32,79,42,90)(33,84,37,89)(34,83,38,88)(35,82,39,87)(36,81,40,86), (1,54,18,43)(2,49,13,44)(3,50,14,45)(4,51,15,46)(5,52,16,47)(6,53,17,48)(7,61,95,56)(8,62,96,57)(9,63,91,58)(10,64,92,59)(11,65,93,60)(12,66,94,55)(19,31,29,41)(20,32,30,42)(21,33,25,37)(22,34,26,38)(23,35,27,39)(24,36,28,40)(67,90,78,79)(68,85,73,80)(69,86,74,81)(70,87,75,82)(71,88,76,83)(72,89,77,84)>;`

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 4F 4G ··· 4N 4O 4P 4Q 4R 6A 6B 6C 6D 6E 6F 12A 12B 12C 12D 12E 12F 12G order 1 2 2 2 2 2 2 3 4 4 4 4 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 2 2 2 4 4 4 4 6 ··· 6 12 12 12 12 2 2 2 4 4 8 4 4 4 4 8 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 D6 D6 D6 D6 C4○D4 C4○D4 2- 1+4 D4⋊2S3 S3×C4○D4 Q8○D12 kernel C6.802- 1+4 C23.16D6 Dic3.D4 C23.8D6 Dic6⋊C4 Dic3.Q8 C4.Dic6 C2×Dic3⋊C4 C23.26D6 D4×Dic3 C23.23D6 C3×C22.D4 C22.D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 Dic3 C2×C6 C6 C22 C2 C2 # reps 1 2 2 2 1 2 1 1 1 1 1 1 1 3 2 1 1 4 4 1 2 2 2

Matrix representation of C6.802- 1+4 in GL6(𝔽13)

 12 12 0 0 0 0 1 0 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 12
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 5 0 0 0 0 8 0 0 0 0 0 0 0 8 0 0 0 0 0 0 8
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 1 12
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 1 0 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 8 10 0 0 0 0 8 5

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

C6.802- 1+4 in GAP, Magma, Sage, TeX

`C_6._{80}2_-^{1+4}`
`% in TeX`

`G:=Group("C6.80ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1209);`
`// by ID`

`G=gap.SmallGroup(192,1209);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,758,100,346,297,6278]);`
`// Polycyclic`

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

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