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## G = C6.782- 1+4order 192 = 26·3

### 33rd 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 — C6.782- 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — C2×C3⋊D4 — Dic3⋊4D4 — C6.782- 1+4
 Lower central C3 — C2×C6 — C6.782- 1+4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for C6.782- 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=b2, e2=a3, bab-1=cac=dad-1=a-1, ae=ea, cbc=a3b-1, bd=db, ebe-1=a3b, dcd-1=a3c, ce=ec, ede-1=a3b2d >

Subgroups: 544 in 218 conjugacy classes, 95 normal (31 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×Q8, C22×S3, C22×C6, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C22×Dic3, C2×C3⋊D4, C22×C12, C6×Q8, C22.33C24, Dic34D4, C23.21D6, C4.Dic6, D6.D4, C4⋊C4⋊S3, C2×Dic3⋊C4, C127D4, D63Q8, C3×C22⋊Q8, C6.782- 1+4
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, 2- 1+4, Q83S3, S3×C23, C22.33C24, D46D6, C2×Q83S3, Q8○D12, C6.782- 1+4

Smallest permutation representation of C6.782- 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 96 17 9)(2 95 18 8)(3 94 13 7)(4 93 14 12)(5 92 15 11)(6 91 16 10)(19 83 30 88)(20 82 25 87)(21 81 26 86)(22 80 27 85)(23 79 28 90)(24 84 29 89)(31 76 42 71)(32 75 37 70)(33 74 38 69)(34 73 39 68)(35 78 40 67)(36 77 41 72)(43 64 54 59)(44 63 49 58)(45 62 50 57)(46 61 51 56)(47 66 52 55)(48 65 53 60)
(2 6)(3 5)(7 95)(8 94)(9 93)(10 92)(11 91)(12 96)(13 15)(16 18)(19 20)(21 24)(22 23)(25 30)(26 29)(27 28)(31 35)(32 34)(37 39)(40 42)(43 44)(45 48)(46 47)(49 54)(50 53)(51 52)(55 64)(56 63)(57 62)(58 61)(59 66)(60 65)(67 73)(68 78)(69 77)(70 76)(71 75)(72 74)(79 88)(80 87)(81 86)(82 85)(83 90)(84 89)
(1 53 17 48)(2 52 18 47)(3 51 13 46)(4 50 14 45)(5 49 15 44)(6 54 16 43)(7 61 94 56)(8 66 95 55)(9 65 96 60)(10 64 91 59)(11 63 92 58)(12 62 93 57)(19 40 30 35)(20 39 25 34)(21 38 26 33)(22 37 27 32)(23 42 28 31)(24 41 29 36)(67 88 78 83)(68 87 73 82)(69 86 74 81)(70 85 75 80)(71 90 76 79)(72 89 77 84)
(1 33 4 36)(2 34 5 31)(3 35 6 32)(7 70 10 67)(8 71 11 68)(9 72 12 69)(13 40 16 37)(14 41 17 38)(15 42 18 39)(19 46 22 43)(20 47 23 44)(21 48 24 45)(25 52 28 49)(26 53 29 50)(27 54 30 51)(55 87 58 90)(56 88 59 85)(57 89 60 86)(61 83 64 80)(62 84 65 81)(63 79 66 82)(73 95 76 92)(74 96 77 93)(75 91 78 94)```

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A ··· 4F 4G 4H 4I 4J 4K 4L 4M 4N 6A 6B 6C 6D 6E 12A 12B 12C 12D 12E 12F 12G 12H order 1 2 2 2 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 2 2 12 12 2 4 ··· 4 6 6 6 6 12 12 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 4 type + + + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 D6 C4○D4 2+ 1+4 2- 1+4 Q8⋊3S3 D4⋊6D6 Q8○D12 kernel C6.782- 1+4 Dic3⋊4D4 C23.21D6 C4.Dic6 D6.D4 C4⋊C4⋊S3 C2×Dic3⋊C4 C12⋊7D4 D6⋊3Q8 C3×C22⋊Q8 C22⋊Q8 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C2×C6 C6 C6 C22 C2 C2 # reps 1 2 2 2 2 2 1 1 2 1 1 2 3 1 1 4 1 1 2 2 2

Matrix representation of C6.782- 1+4 in GL8(𝔽13)

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

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

C6.782- 1+4 in GAP, Magma, Sage, TeX

`C_6._{78}2_-^{1+4}`
`% in TeX`

`G:=Group("C6.78ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1204);`
`// by ID`

`G=gap.SmallGroup(192,1204);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,387,100,675,409,80,6278]);`
`// Polycyclic`

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

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