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## G = C6.202- 1+4order 192 = 26·3

### 20th non-split extension by C6 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C6.202- 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — S3×C4⋊C4 — C6.202- 1+4
 Lower central C3 — C2×C6 — C6.202- 1+4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for C6.202- 1+4
G = < a,b,c,d,e | a6=b4=1, c2=a3, d2=b2, e2=a3b2, ab=ba, cac-1=dad-1=a-1, ae=ea, cbc-1=a3b-1, bd=db, ebe-1=a3b, cd=dc, ece-1=a3c, ede-1=a3b2d >

Subgroups: 544 in 218 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4×S3, D12, C2×Dic3, C3⋊D4, 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, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C22×C12, C6×Q8, C22.33C24, C23.8D6, C23.9D6, Dic3⋊D4, Dic3.Q8, S3×C4⋊C4, Dic35D4, D6.D4, C4⋊C4⋊S3, C4×C3⋊D4, C23.28D6, D63Q8, C3×C22⋊Q8, C6.202- 1+4
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, 2- 1+4, S3×C23, C22.33C24, D46D6, Q8.15D6, S3×C4○D4, C6.202- 1+4

Smallest permutation representation of C6.202- 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 67 10 64)(2 68 11 65)(3 69 12 66)(4 70 7 61)(5 71 8 62)(6 72 9 63)(13 55 22 52)(14 56 23 53)(15 57 24 54)(16 58 19 49)(17 59 20 50)(18 60 21 51)(25 91 34 88)(26 92 35 89)(27 93 36 90)(28 94 31 85)(29 95 32 86)(30 96 33 87)(37 79 46 76)(38 80 47 77)(39 81 48 78)(40 82 43 73)(41 83 44 74)(42 84 45 75)
(1 34 4 31)(2 33 5 36)(3 32 6 35)(7 28 10 25)(8 27 11 30)(9 26 12 29)(13 37 16 40)(14 42 17 39)(15 41 18 38)(19 43 22 46)(20 48 23 45)(21 47 24 44)(49 79 52 82)(50 84 53 81)(51 83 54 80)(55 73 58 76)(56 78 59 75)(57 77 60 74)(61 88 64 85)(62 87 65 90)(63 86 66 89)(67 94 70 91)(68 93 71 96)(69 92 72 95)
(1 61 10 70)(2 66 11 69)(3 65 12 68)(4 64 7 67)(5 63 8 72)(6 62 9 71)(13 58 22 49)(14 57 23 54)(15 56 24 53)(16 55 19 52)(17 60 20 51)(18 59 21 50)(25 91 34 88)(26 96 35 87)(27 95 36 86)(28 94 31 85)(29 93 32 90)(30 92 33 89)(37 76 46 79)(38 75 47 84)(39 74 48 83)(40 73 43 82)(41 78 44 81)(42 77 45 80)
(1 55 7 49)(2 56 8 50)(3 57 9 51)(4 58 10 52)(5 59 11 53)(6 60 12 54)(13 67 19 61)(14 68 20 62)(15 69 21 63)(16 70 22 64)(17 71 23 65)(18 72 24 66)(25 79 31 73)(26 80 32 74)(27 81 33 75)(28 82 34 76)(29 83 35 77)(30 84 36 78)(37 91 43 85)(38 92 44 86)(39 93 45 87)(40 94 46 88)(41 95 47 89)(42 96 48 90)```

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C ··· 4G 4H 4I 4J ··· 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 6 6 6 6 6 12 12 12 12 12 12 12 12 size 1 1 1 1 4 6 6 12 2 2 2 4 ··· 4 6 6 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 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 C2 C2 C2 S3 D6 D6 D6 D6 C4○D4 2+ 1+4 2- 1+4 D4⋊6D6 Q8.15D6 S3×C4○D4 kernel C6.202- 1+4 C23.8D6 C23.9D6 Dic3⋊D4 Dic3.Q8 S3×C4⋊C4 Dic3⋊5D4 D6.D4 C4⋊C4⋊S3 C4×C3⋊D4 C23.28D6 D6⋊3Q8 C3×C22⋊Q8 C22⋊Q8 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 D6 C6 C6 C2 C2 C2 # reps 1 1 2 1 2 1 1 1 1 1 1 2 1 1 2 3 1 1 4 1 1 2 2 2

Matrix representation of C6.202- 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
,
 8 0 0 0 0 0 6 5 0 0 0 0 0 0 10 7 6 12 0 0 6 3 1 7 0 0 0 0 3 6 0 0 0 0 7 10
,
 5 0 0 0 0 0 7 8 0 0 0 0 0 0 12 0 2 0 0 0 1 1 11 11 0 0 0 0 1 0 0 0 0 0 12 12
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 3 6 7 1 0 0 3 10 7 6 0 0 3 6 10 7 0 0 3 10 10 3
,
 4 11 0 0 0 0 1 9 0 0 0 0 0 0 8 0 10 0 0 0 0 8 0 10 0 0 0 0 5 0 0 0 0 0 0 5

`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],[8,6,0,0,0,0,0,5,0,0,0,0,0,0,10,6,0,0,0,0,7,3,0,0,0,0,6,1,3,7,0,0,12,7,6,10],[5,7,0,0,0,0,0,8,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,2,11,1,12,0,0,0,11,0,12],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,3,3,3,3,0,0,6,10,6,10,0,0,7,7,10,10,0,0,1,6,7,3],[4,1,0,0,0,0,11,9,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,10,0,5,0,0,0,0,10,0,5] >;`

C6.202- 1+4 in GAP, Magma, Sage, TeX

`C_6._{20}2_-^{1+4}`
`% in TeX`

`G:=Group("C6.20ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1197);`
`// by ID`

`G=gap.SmallGroup(192,1197);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,268,675,570,6278]);`
`// Polycyclic`

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

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