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## G = C6.222- 1+4order 192 = 26·3

### 22nd 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 — C6.222- 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — D6⋊Q8 — C6.222- 1+4
 Lower central C3 — C2×C6 — C6.222- 1+4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for C6.222- 1+4
G = < a,b,c,d,e | a6=b4=e2=1, c2=a3, d2=a3b2, bab-1=dad-1=a-1, ac=ca, ae=ea, cbc-1=a3b-1, bd=db, ebe=a3b, cd=dc, ce=ec, ede=a3b2d >

Subgroups: 544 in 216 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, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×Q8, C22×S3, C22×C6, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C422C2, C4⋊Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C2×C3⋊D4, C22×C12, C6×Q8, C22.36C24, C23.8D6, Dic3⋊D4, C23.11D6, Dic6⋊C4, C4⋊C47S3, D6.D4, D6⋊Q8, C4⋊C4⋊S3, C4×C3⋊D4, C23.28D6, Dic3⋊Q8, C12.23D4, C3×C22⋊Q8, C6.222- 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, C22.36C24, D46D6, Q8.15D6, S3×C4○D4, C6.222- 1+4

Smallest permutation representation of C6.222- 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 48 17 53)(2 47 18 52)(3 46 13 51)(4 45 14 50)(5 44 15 49)(6 43 16 54)(7 64 94 59)(8 63 95 58)(9 62 96 57)(10 61 91 56)(11 66 92 55)(12 65 93 60)(19 37 30 32)(20 42 25 31)(21 41 26 36)(22 40 27 35)(23 39 28 34)(24 38 29 33)(67 83 78 88)(68 82 73 87)(69 81 74 86)(70 80 75 85)(71 79 76 90)(72 84 77 89)
(1 57 4 60)(2 58 5 55)(3 59 6 56)(7 51 10 54)(8 52 11 49)(9 53 12 50)(13 64 16 61)(14 65 17 62)(15 66 18 63)(19 70 22 67)(20 71 23 68)(21 72 24 69)(25 76 28 73)(26 77 29 74)(27 78 30 75)(31 82 34 79)(32 83 35 80)(33 84 36 81)(37 88 40 85)(38 89 41 86)(39 90 42 87)(43 94 46 91)(44 95 47 92)(45 96 48 93)
(1 26 14 24)(2 25 15 23)(3 30 16 22)(4 29 17 21)(5 28 18 20)(6 27 13 19)(7 85 91 83)(8 90 92 82)(9 89 93 81)(10 88 94 80)(11 87 95 79)(12 86 96 84)(31 49 39 47)(32 54 40 46)(33 53 41 45)(34 52 42 44)(35 51 37 43)(36 50 38 48)(55 73 63 71)(56 78 64 70)(57 77 65 69)(58 76 66 68)(59 75 61 67)(60 74 62 72)
(1 24)(2 19)(3 20)(4 21)(5 22)(6 23)(7 82)(8 83)(9 84)(10 79)(11 80)(12 81)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(31 54)(32 49)(33 50)(34 51)(35 52)(36 53)(37 44)(38 45)(39 46)(40 47)(41 48)(42 43)(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 92)(86 93)(87 94)(88 95)(89 96)(90 91)```

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C ··· 4G 4H 4I 4J 4K 4L 4M 4N 4O 6A 6B 6C 6D 6E 12A 12B 12C 12D 12E 12F 12G 12H 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 12 12 12 12 12 12 12 12 size 1 1 1 1 4 12 12 2 2 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 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 C2 S3 D6 D6 D6 D6 C4○D4 2+ 1+4 2- 1+4 D4⋊6D6 Q8.15D6 S3×C4○D4 kernel C6.222- 1+4 C23.8D6 Dic3⋊D4 C23.11D6 Dic6⋊C4 C4⋊C4⋊7S3 D6.D4 D6⋊Q8 C4⋊C4⋊S3 C4×C3⋊D4 C23.28D6 Dic3⋊Q8 C12.23D4 C3×C22⋊Q8 C22⋊Q8 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 Dic3 C6 C6 C2 C2 C2 # reps 1 1 1 2 1 1 1 2 1 1 1 1 1 1 1 2 3 1 1 4 1 1 2 2 2

Matrix representation of C6.222- 1+4 in GL8(𝔽13)

 1 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 12 0 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
,
 4 8 8 3 0 0 0 0 4 9 8 5 0 0 0 0 5 10 9 5 0 0 0 0 5 8 9 4 0 0 0 0 0 0 0 0 0 2 1 2 0 0 0 0 0 10 10 10 0 0 0 0 12 8 0 0 0 0 0 0 1 12 3 3
,
 5 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 10 0 5
,
 0 0 1 0 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 0 0 0 0 0 0 1 5 0 0 0 0 0 0 10 12 0 0 0 0 0 0 0 2 1 2 0 0 0 0 3 0 12 12
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 5 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 2 1 2 0 0 0 0 0 0 0 12

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

C6.222- 1+4 in GAP, Magma, Sage, TeX

`C_6._{22}2_-^{1+4}`
`% in TeX`

`G:=Group("C6.22ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1199);`
`// by ID`

`G=gap.SmallGroup(192,1199);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,555,100,1571,297,6278]);`
`// Polycyclic`

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

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