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## G = C6.1442+ 1+4order 192 = 26·3

### 53rd 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 — C6 — C6.1442+ 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — D4×Dic3 — C6.1442+ 1+4
 Lower central C3 — C6 — C6.1442+ 1+4
 Upper central C1 — C22 — C2×C4○D4

Generators and relations for C6.1442+ 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=a3b2, e2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 552 in 294 conjugacy classes, 191 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C2×C4○D4, C4×Dic3, C4⋊Dic3, C4⋊Dic3, C6.D4, C22×Dic3, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C23.33C23, C2×C4⋊Dic3, C23.26D6, D4×Dic3, Q8×Dic3, C6×C4○D4, C6.1442+ 1+4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C24, C2×Dic3, C22×S3, C23×C4, 2+ 1+4, 2- 1+4, C22×Dic3, S3×C23, C23.33C23, D4○D12, Q8○D12, C23×Dic3, C6.1442+ 1+4

Smallest permutation representation of C6.1442+ 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 19 18 29)(2 20 13 30)(3 21 14 25)(4 22 15 26)(5 23 16 27)(6 24 17 28)(7 80 95 85)(8 81 96 86)(9 82 91 87)(10 83 92 88)(11 84 93 89)(12 79 94 90)(31 46 41 51)(32 47 42 52)(33 48 37 53)(34 43 38 54)(35 44 39 49)(36 45 40 50)(55 75 66 70)(56 76 61 71)(57 77 62 72)(58 78 63 67)(59 73 64 68)(60 74 65 69)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 56)(8 57)(9 58)(10 59)(11 60)(12 55)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 34)(20 35)(21 36)(22 31)(23 32)(24 33)(25 40)(26 41)(27 42)(28 37)(29 38)(30 39)(61 95)(62 96)(63 91)(64 92)(65 93)(66 94)(67 82)(68 83)(69 84)(70 79)(71 80)(72 81)(73 88)(74 89)(75 90)(76 85)(77 86)(78 87)
(1 59 15 61)(2 58 16 66)(3 57 17 65)(4 56 18 64)(5 55 13 63)(6 60 14 62)(7 54 92 46)(8 53 93 45)(9 52 94 44)(10 51 95 43)(11 50 96 48)(12 49 91 47)(19 73 26 71)(20 78 27 70)(21 77 28 69)(22 76 29 68)(23 75 30 67)(24 74 25 72)(31 85 38 83)(32 90 39 82)(33 89 40 81)(34 88 41 80)(35 87 42 79)(36 86 37 84)
(1 22 18 26)(2 23 13 27)(3 24 14 28)(4 19 15 29)(5 20 16 30)(6 21 17 25)(7 83 95 88)(8 84 96 89)(9 79 91 90)(10 80 92 85)(11 81 93 86)(12 82 94 87)(31 54 41 43)(32 49 42 44)(33 50 37 45)(34 51 38 46)(35 52 39 47)(36 53 40 48)(55 67 66 78)(56 68 61 73)(57 69 62 74)(58 70 63 75)(59 71 64 76)(60 72 65 77)```

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 3 4A ··· 4H 4I ··· 4X 6A 6B 6C 6D ··· 6I 12A 12B 12C 12D 12E ··· 12J order 1 2 2 2 2 ··· 2 3 4 ··· 4 4 ··· 4 6 6 6 6 ··· 6 12 12 12 12 12 ··· 12 size 1 1 1 1 2 ··· 2 2 2 ··· 2 6 ··· 6 2 2 2 4 ··· 4 2 2 2 2 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + - + - + - image C1 C2 C2 C2 C2 C2 C4 S3 D6 D6 D6 Dic3 2+ 1+4 2- 1+4 D4○D12 Q8○D12 kernel C6.1442+ 1+4 C2×C4⋊Dic3 C23.26D6 D4×Dic3 Q8×Dic3 C6×C4○D4 C3×C4○D4 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 C6 C6 C2 C2 # reps 1 3 3 6 2 1 16 1 3 3 1 8 1 1 2 2

Matrix representation of C6.1442+ 1+4 in GL6(𝔽13)

 0 1 0 0 0 0 12 1 0 0 0 0 0 0 0 1 0 0 0 0 12 1 0 0 0 0 0 0 0 1 0 0 0 0 12 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 3 7 0 0 0 0 6 10 0 0 0 0 10 6 10 6 0 0 7 3 7 3
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 3 7 6 1 0 0 6 10 12 7 0 0 10 6 10 6 0 0 7 3 7 3
,
 0 8 0 0 0 0 8 0 0 0 0 0 0 0 0 12 0 11 0 0 12 0 11 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 10 6 0 0 0 0 7 3 0 0 0 0 0 0 10 6 0 0 0 0 7 3

`G:=sub<GL(6,GF(13))| [0,12,0,0,0,0,1,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,6,10,7,0,0,7,10,6,3,0,0,0,0,10,7,0,0,0,0,6,3],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,6,10,7,0,0,7,10,6,3,0,0,6,12,10,7,0,0,1,7,6,3],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,11,0,1,0,0,11,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,10,7,0,0,0,0,6,3,0,0,0,0,0,0,10,7,0,0,0,0,6,3] >;`

C6.1442+ 1+4 in GAP, Magma, Sage, TeX

`C_6._{144}2_+^{1+4}`
`% in TeX`

`G:=Group("C6.144ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1386);`
`// by ID`

`G=gap.SmallGroup(192,1386);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,387,1123,136,6278]);`
`// Polycyclic`

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