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## G = C6.642+ 1+4order 192 = 26·3

### 64th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C6.642+ 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — S3×C4⋊C4 — C6.642+ 1+4
 Lower central C3 — C2×C6 — C6.642+ 1+4
 Upper central C1 — C22 — C22.D4

Generators and relations for C6.642+ 1+4
G = < a,b,c,d,e | a6=b4=e2=1, c2=a3, d2=a3b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, dbd-1=ebe=a3b, cd=dc, ce=ec, ede=a3b2d >

Subgroups: 608 in 238 conjugacy classes, 95 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C2×C4⋊C4, C42⋊C2, C4×D4, C4⋊D4, C22.D4, 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, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, C22.47C24, C23.16D6, C23.8D6, Dic34D4, C23.9D6, Dic3⋊D4, Dic3.Q8, S3×C4⋊C4, Dic35D4, C4⋊C4⋊S3, C4×C3⋊D4, D63D4, C23.14D6, C3×C22.D4, C6.642+ 1+4
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, S3×C23, C22.47C24, D46D6, S3×C4○D4, C6.642+ 1+4

Smallest permutation representation of C6.642+ 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 82 7 76)(2 83 8 77)(3 84 9 78)(4 79 10 73)(5 80 11 74)(6 81 12 75)(13 94 19 88)(14 95 20 89)(15 96 21 90)(16 91 22 85)(17 92 23 86)(18 93 24 87)(25 55 31 49)(26 56 32 50)(27 57 33 51)(28 58 34 52)(29 59 35 53)(30 60 36 54)(37 67 43 61)(38 68 44 62)(39 69 45 63)(40 70 46 64)(41 71 47 65)(42 72 48 66)
(1 52 4 49)(2 53 5 50)(3 54 6 51)(7 58 10 55)(8 59 11 56)(9 60 12 57)(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 79 10 76)(2 84 11 75)(3 83 12 74)(4 82 7 73)(5 81 8 78)(6 80 9 77)(13 85 22 94)(14 90 23 93)(15 89 24 92)(16 88 19 91)(17 87 20 96)(18 86 21 95)(25 49 34 58)(26 54 35 57)(27 53 36 56)(28 52 31 55)(29 51 32 60)(30 50 33 59)(37 67 46 64)(38 72 47 63)(39 71 48 62)(40 70 43 61)(41 69 44 66)(42 68 45 65)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 40)(26 41)(27 42)(28 37)(29 38)(30 39)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(73 88)(74 89)(75 90)(76 85)(77 86)(78 87)(79 94)(80 95)(81 96)(82 91)(83 92)(84 93)```

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 3 4A 4B 4C 4D 4E 4F 4G 4H ··· 4M 4N 4O 4P 6A 6B 6C 6D 6E 6F 12A 12B 12C 12D 12E 12F 12G order 1 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 ··· 4 4 4 4 6 6 6 6 6 6 12 12 12 12 12 12 12 size 1 1 1 1 4 4 6 6 12 2 2 2 2 2 4 4 4 6 ··· 6 12 12 12 2 2 2 4 4 8 4 4 4 4 8 8 8

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 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 C4○D4 2+ 1+4 D4⋊6D6 S3×C4○D4 kernel C6.642+ 1+4 C23.16D6 C23.8D6 Dic3⋊4D4 C23.9D6 Dic3⋊D4 Dic3.Q8 S3×C4⋊C4 Dic3⋊5D4 C4⋊C4⋊S3 C4×C3⋊D4 D6⋊3D4 C23.14D6 C3×C22.D4 C22.D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 Dic3 D6 C6 C2 C2 # reps 1 1 1 1 1 2 1 1 1 1 2 1 1 1 1 3 2 1 1 4 4 1 2 4

Matrix representation of C6.642+ 1+4 in GL6(𝔽13)

 0 12 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 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 11 0 0 0 0 1 1 0 0 0 0 0 0 0 5 0 0 0 0 8 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 5 10 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 1 2 0 0 0 0 0 12 0 0 0 0 0 0 0 5 0 0 0 0 5 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 11 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1

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

C6.642+ 1+4 in GAP, Magma, Sage, TeX

`C_6._{64}2_+^{1+4}`
`% in TeX`

`G:=Group("C6.64ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1220);`
`// by ID`

`G=gap.SmallGroup(192,1220);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,219,184,1571,297,6278]);`
`// 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,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=b^-1,d*b*d^-1=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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