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## G = C6.722- 1+4order 192 = 26·3

### 27th 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 — C6.722- 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C2×C4○D12 — C6.722- 1+4
 Lower central C3 — C2×C6 — C6.722- 1+4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C6.722- 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=b-1, dbd-1=ebe-1=a3b, dcd-1=ece-1=a3c, ede-1=b2d >

Subgroups: 784 in 294 conjugacy classes, 103 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C2×C4⋊C4, C4⋊D4, C4⋊D4, C22⋊Q8, C2×C4○D4, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, C4○D12, D42S3, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, C22.31C24, Dic3.D4, Dic3⋊D4, S3×C4⋊C4, C4.D12, C12.48D4, D63D4, D63D4, C23.14D6, C3×C4⋊D4, C2×C4○D12, C2×D42S3, C6.722- 1+4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, 2+ 1+4, 2- 1+4, S3×D4, S3×C23, C22.31C24, C2×S3×D4, D46D6, Q8○D12, C6.722- 1+4

Smallest permutation representation of C6.722- 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 91 18 9)(2 96 13 8)(3 95 14 7)(4 94 15 12)(5 93 16 11)(6 92 17 10)(19 79 29 89)(20 84 30 88)(21 83 25 87)(22 82 26 86)(23 81 27 85)(24 80 28 90)(31 77 41 67)(32 76 42 72)(33 75 37 71)(34 74 38 70)(35 73 39 69)(36 78 40 68)(43 58 53 62)(44 57 54 61)(45 56 49 66)(46 55 50 65)(47 60 51 64)(48 59 52 63)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 25)(7 90)(8 85)(9 86)(10 87)(11 88)(12 89)(13 23)(14 24)(15 19)(16 20)(17 21)(18 22)(31 50)(32 51)(33 52)(34 53)(35 54)(36 49)(37 48)(38 43)(39 44)(40 45)(41 46)(42 47)(55 77)(56 78)(57 73)(58 74)(59 75)(60 76)(61 69)(62 70)(63 71)(64 72)(65 67)(66 68)(79 94)(80 95)(81 96)(82 91)(83 92)(84 93)
(1 70 15 77)(2 69 16 76)(3 68 17 75)(4 67 18 74)(5 72 13 73)(6 71 14 78)(7 37 92 36)(8 42 93 35)(9 41 94 34)(10 40 95 33)(11 39 96 32)(12 38 91 31)(19 58 26 65)(20 57 27 64)(21 56 28 63)(22 55 29 62)(23 60 30 61)(24 59 25 66)(43 79 50 86)(44 84 51 85)(45 83 52 90)(46 82 53 89)(47 81 54 88)(48 80 49 87)
(1 91 15 12)(2 92 16 7)(3 93 17 8)(4 94 18 9)(5 95 13 10)(6 96 14 11)(19 86 26 79)(20 87 27 80)(21 88 28 81)(22 89 29 82)(23 90 30 83)(24 85 25 84)(31 67 38 74)(32 68 39 75)(33 69 40 76)(34 70 41 77)(35 71 42 78)(36 72 37 73)(43 55 50 62)(44 56 51 63)(45 57 52 64)(46 58 53 65)(47 59 54 66)(48 60 49 61)```

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G 4H ··· 4L 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D 12E 12F order 1 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 ··· 4 6 6 6 6 6 6 6 12 12 12 12 12 12 size 1 1 1 1 4 4 4 6 6 12 2 2 2 4 4 4 6 6 12 ··· 12 2 2 2 4 4 8 8 4 4 4 4 8 8

36 irreducible representations

 dim 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 S3 D4 D6 D6 D6 D6 2+ 1+4 2- 1+4 S3×D4 D4⋊6D6 Q8○D12 kernel C6.722- 1+4 Dic3.D4 Dic3⋊D4 S3×C4⋊C4 C4.D12 C12.48D4 D6⋊3D4 C23.14D6 C3×C4⋊D4 C2×C4○D12 C2×D4⋊2S3 C4⋊D4 C4×S3 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C6 C6 C4 C2 C2 # reps 1 2 2 1 1 1 3 2 1 1 1 1 4 2 1 1 3 1 1 2 2 2

Matrix representation of C6.722- 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
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 0 0 3 6 0 0 0 0 3 10 0 0 10 7 0 0 0 0 10 3 0 0
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 4 8 4 0 0 0 4 9 9 9 0 0 4 0 4 8 0 0 9 9 4 9
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 0 0 3 6 0 0 0 0 7 10 0 0 3 6 0 0 0 0 7 10 0 0

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

C6.722- 1+4 in GAP, Magma, Sage, TeX

`C_6._{72}2_-^{1+4}`
`% in TeX`

`G:=Group("C6.72ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1167);`
`// by ID`

`G=gap.SmallGroup(192,1167);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,1123,570,185,6278]);`
`// 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=b^-1,d*b*d^-1=e*b*e^-1=a^3*b,d*c*d^-1=e*c*e^-1=a^3*c,e*d*e^-1=b^2*d>;`
`// generators/relations`

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