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## G = C6.212- 1+4order 192 = 26·3

### 21st 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 — C6.212- 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — S3×C4⋊C4 — C6.212- 1+4
 Lower central C3 — C2×C6 — C6.212- 1+4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for C6.212- 1+4
G = < a,b,c,d,e | a6=b4=1, c2=a3, d2=a3b2, e2=b2, ab=ba, cac-1=dad-1=a-1, ae=ea, cbc-1=a3b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 464 in 214 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, D6, 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, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×Q8, C22×S3, C22×C6, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, C4×Dic3, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, C2×C3⋊D4, C22×C12, C6×Q8, C22.46C24, C23.8D6, C23.9D6, Dic3.Q8, C4.Dic6, S3×C4⋊C4, C4⋊C47S3, C23.26D6, C4×C3⋊D4, Q8×Dic3, D63Q8, C3×C22⋊Q8, C6.212- 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, Q8.15D6, S3×C4○D4, C6.212- 1+4

Smallest permutation representation of C6.212- 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 70 7 64)(2 71 8 65)(3 72 9 66)(4 67 10 61)(5 68 11 62)(6 69 12 63)(13 58 19 52)(14 59 20 53)(15 60 21 54)(16 55 22 49)(17 56 23 50)(18 57 24 51)(25 91 31 85)(26 92 32 86)(27 93 33 87)(28 94 34 88)(29 95 35 89)(30 96 36 90)(37 79 43 73)(38 80 44 74)(39 81 45 75)(40 82 46 76)(41 83 47 77)(42 84 48 78)
(1 64 4 61)(2 63 5 66)(3 62 6 65)(7 70 10 67)(8 69 11 72)(9 68 12 71)(13 55 16 58)(14 60 17 57)(15 59 18 56)(19 49 22 52)(20 54 23 51)(21 53 24 50)(25 85 28 88)(26 90 29 87)(27 89 30 86)(31 91 34 94)(32 96 35 93)(33 95 36 92)(37 82 40 79)(38 81 41 84)(39 80 42 83)(43 76 46 73)(44 75 47 78)(45 74 48 77)
(1 28 10 31)(2 27 11 36)(3 26 12 35)(4 25 7 34)(5 30 8 33)(6 29 9 32)(13 43 22 40)(14 48 23 39)(15 47 24 38)(16 46 19 37)(17 45 20 42)(18 44 21 41)(49 82 58 73)(50 81 59 78)(51 80 60 77)(52 79 55 76)(53 84 56 75)(54 83 57 74)(61 85 70 94)(62 90 71 93)(63 89 72 92)(64 88 67 91)(65 87 68 96)(66 86 69 95)
(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), (1,70,7,64)(2,71,8,65)(3,72,9,66)(4,67,10,61)(5,68,11,62)(6,69,12,63)(13,58,19,52)(14,59,20,53)(15,60,21,54)(16,55,22,49)(17,56,23,50)(18,57,24,51)(25,91,31,85)(26,92,32,86)(27,93,33,87)(28,94,34,88)(29,95,35,89)(30,96,36,90)(37,79,43,73)(38,80,44,74)(39,81,45,75)(40,82,46,76)(41,83,47,77)(42,84,48,78), (1,64,4,61)(2,63,5,66)(3,62,6,65)(7,70,10,67)(8,69,11,72)(9,68,12,71)(13,55,16,58)(14,60,17,57)(15,59,18,56)(19,49,22,52)(20,54,23,51)(21,53,24,50)(25,85,28,88)(26,90,29,87)(27,89,30,86)(31,91,34,94)(32,96,35,93)(33,95,36,92)(37,82,40,79)(38,81,41,84)(39,80,42,83)(43,76,46,73)(44,75,47,78)(45,74,48,77), (1,28,10,31)(2,27,11,36)(3,26,12,35)(4,25,7,34)(5,30,8,33)(6,29,9,32)(13,43,22,40)(14,48,23,39)(15,47,24,38)(16,46,19,37)(17,45,20,42)(18,44,21,41)(49,82,58,73)(50,81,59,78)(51,80,60,77)(52,79,55,76)(53,84,56,75)(54,83,57,74)(61,85,70,94)(62,90,71,93)(63,89,72,92)(64,88,67,91)(65,87,68,96)(66,86,69,95), (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), (1,70,7,64)(2,71,8,65)(3,72,9,66)(4,67,10,61)(5,68,11,62)(6,69,12,63)(13,58,19,52)(14,59,20,53)(15,60,21,54)(16,55,22,49)(17,56,23,50)(18,57,24,51)(25,91,31,85)(26,92,32,86)(27,93,33,87)(28,94,34,88)(29,95,35,89)(30,96,36,90)(37,79,43,73)(38,80,44,74)(39,81,45,75)(40,82,46,76)(41,83,47,77)(42,84,48,78), (1,64,4,61)(2,63,5,66)(3,62,6,65)(7,70,10,67)(8,69,11,72)(9,68,12,71)(13,55,16,58)(14,60,17,57)(15,59,18,56)(19,49,22,52)(20,54,23,51)(21,53,24,50)(25,85,28,88)(26,90,29,87)(27,89,30,86)(31,91,34,94)(32,96,35,93)(33,95,36,92)(37,82,40,79)(38,81,41,84)(39,80,42,83)(43,76,46,73)(44,75,47,78)(45,74,48,77), (1,28,10,31)(2,27,11,36)(3,26,12,35)(4,25,7,34)(5,30,8,33)(6,29,9,32)(13,43,22,40)(14,48,23,39)(15,47,24,38)(16,46,19,37)(17,45,20,42)(18,44,21,41)(49,82,58,73)(50,81,59,78)(51,80,60,77)(52,79,55,76)(53,84,56,75)(54,83,57,74)(61,85,70,94)(62,90,71,93)(63,89,72,92)(64,88,67,91)(65,87,68,96)(66,86,69,95), (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)], [(1,70,7,64),(2,71,8,65),(3,72,9,66),(4,67,10,61),(5,68,11,62),(6,69,12,63),(13,58,19,52),(14,59,20,53),(15,60,21,54),(16,55,22,49),(17,56,23,50),(18,57,24,51),(25,91,31,85),(26,92,32,86),(27,93,33,87),(28,94,34,88),(29,95,35,89),(30,96,36,90),(37,79,43,73),(38,80,44,74),(39,81,45,75),(40,82,46,76),(41,83,47,77),(42,84,48,78)], [(1,64,4,61),(2,63,5,66),(3,62,6,65),(7,70,10,67),(8,69,11,72),(9,68,12,71),(13,55,16,58),(14,60,17,57),(15,59,18,56),(19,49,22,52),(20,54,23,51),(21,53,24,50),(25,85,28,88),(26,90,29,87),(27,89,30,86),(31,91,34,94),(32,96,35,93),(33,95,36,92),(37,82,40,79),(38,81,41,84),(39,80,42,83),(43,76,46,73),(44,75,47,78),(45,74,48,77)], [(1,28,10,31),(2,27,11,36),(3,26,12,35),(4,25,7,34),(5,30,8,33),(6,29,9,32),(13,43,22,40),(14,48,23,39),(15,47,24,38),(16,46,19,37),(17,45,20,42),(18,44,21,41),(49,82,58,73),(50,81,59,78),(51,80,60,77),(52,79,55,76),(53,84,56,75),(54,83,57,74),(61,85,70,94),(62,90,71,93),(63,89,72,92),(64,88,67,91),(65,87,68,96),(66,86,69,95)], [(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)]])`

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4N 4O 4P 4Q 4R 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 4 4 6 6 6 6 6 12 12 12 12 12 12 12 12 size 1 1 1 1 4 6 6 2 2 2 2 2 4 4 4 4 6 ··· 6 12 12 12 12 2 2 2 4 4 4 4 4 4 8 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 Q8.15D6 S3×C4○D4 kernel C6.212- 1+4 C23.8D6 C23.9D6 Dic3.Q8 C4.Dic6 S3×C4⋊C4 C4⋊C4⋊7S3 C23.26D6 C4×C3⋊D4 Q8×Dic3 D6⋊3Q8 C3×C22⋊Q8 C22⋊Q8 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C12 D6 C6 C4 C2 C2 # reps 1 2 2 2 1 1 2 1 1 1 1 1 1 2 3 1 1 4 4 1 2 2 2

Matrix representation of C6.212- 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 0 0 0 0 0 0 12
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 5 0 0 0 0 0 0 5
,
 0 12 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 8 0 0 0 0 0 0 8
,
 5 0 0 0 0 0 0 5 0 0 0 0 0 0 12 0 0 0 0 0 1 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1
,
 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 8 0 0 0 0 8 0

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

C6.212- 1+4 in GAP, Magma, Sage, TeX

`C_6._{21}2_-^{1+4}`
`% in TeX`

`G:=Group("C6.21ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1198);`
`// by ID`

`G=gap.SmallGroup(192,1198);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,268,1571,297,6278]);`
`// Polycyclic`

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

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