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G = C6.52- 1+4order 192 = 26·3

5th 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 — C6.52- 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C4⋊C4⋊7S3 — C6.52- 1+4
 Lower central C3 — C2×C6 — C6.52- 1+4
 Upper central C1 — C22 — C2×C4⋊C4

Generators and relations for C6.52- 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=b2, e2=a3b2, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc=b-1, dbd-1=a3b, be=eb, cd=dc, ce=ec, ede-1=a3b2d >

Subgroups: 456 in 214 conjugacy classes, 99 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C22×S3, C22×C6, C2×C4⋊C4, C42⋊C2, C4×D4, C4×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×C4⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×C3⋊D4, C22×C12, C22×C12, C22.46C24, Dic6⋊C4, Dic3.Q8, C4.Dic6, C4⋊C47S3, C4.D12, C4⋊C4⋊S3, C12.48D4, C23.26D6, C4×C3⋊D4, C23.28D6, C6×C4⋊C4, C6.52- 1+4
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2- 1+4, C4○D12, D42S3, S3×C23, C22.46C24, C2×C4○D12, C2×D42S3, Q8.15D6, C6.52- 1+4

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

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A ··· 4F 4G 4H 4I 4J 4K 4L 4M 4N ··· 4R 6A ··· 6G 12A ··· 12L order 1 2 2 2 2 2 2 3 4 ··· 4 4 4 4 4 4 4 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 12 2 2 ··· 2 4 4 4 6 6 6 6 12 ··· 12 2 ··· 2 4 ··· 4

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D6 D6 C4○D4 C4○D4 C4○D12 2- 1+4 D4⋊2S3 Q8.15D6 kernel C6.52- 1+4 Dic6⋊C4 Dic3.Q8 C4.Dic6 C4⋊C4⋊7S3 C4.D12 C4⋊C4⋊S3 C12.48D4 C23.26D6 C4×C3⋊D4 C23.28D6 C6×C4⋊C4 C2×C4⋊C4 C4⋊C4 C22×C4 C12 C2×C6 C4 C6 C22 C2 # reps 1 1 2 1 1 1 2 1 2 1 2 1 1 4 3 4 4 8 1 2 2

Matrix representation of C6.52- 1+4 in GL4(𝔽13) generated by

 1 1 0 0 12 0 0 0 0 0 12 0 0 0 0 12
,
 2 4 0 0 2 11 0 0 0 0 1 0 0 0 1 12
,
 11 9 0 0 4 2 0 0 0 0 12 0 0 0 0 12
,
 3 6 0 0 7 10 0 0 0 0 8 10 0 0 8 5
,
 1 0 0 0 0 1 0 0 0 0 5 0 0 0 5 8
`G:=sub<GL(4,GF(13))| [1,12,0,0,1,0,0,0,0,0,12,0,0,0,0,12],[2,2,0,0,4,11,0,0,0,0,1,1,0,0,0,12],[11,4,0,0,9,2,0,0,0,0,12,0,0,0,0,12],[3,7,0,0,6,10,0,0,0,0,8,8,0,0,10,5],[1,0,0,0,0,1,0,0,0,0,5,5,0,0,0,8] >;`

C6.52- 1+4 in GAP, Magma, Sage, TeX

`C_6._52_-^{1+4}`
`% in TeX`

`G:=Group("C6.5ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1072);`
`// by ID`

`G=gap.SmallGroup(192,1072);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,387,100,675,6278]);`
`// Polycyclic`

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

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