Copied to
clipboard

## G = C6.452- 1+4order 192 = 26·3

### 45th 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 — C6.452- 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C2×Q8⋊3S3 — C6.452- 1+4
 Lower central C3 — C2×C6 — C6.452- 1+4
 Upper central C1 — C22 — C22×Q8

Generators and relations for C6.452- 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=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: 696 in 290 conjugacy classes, 115 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×S3, C22×C6, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C4.4D4, C22×Q8, C2×C4○D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, S3×C2×C4, C2×D12, Q83S3, C2×C3⋊D4, C22×C12, C6×Q8, C6×Q8, C6×Q8, Q85D4, C4×C3⋊D4, C127D4, Q8×Dic3, D63Q8, C12.23D4, C2×Q83S3, Q8×C2×C6, C6.452- 1+4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C3⋊D4, C22×S3, C22×D4, C2×C4○D4, 2- 1+4, Q83S3, C2×C3⋊D4, S3×C23, Q85D4, C2×Q83S3, Q8.15D6, C22×C3⋊D4, C6.452- 1+4

Smallest permutation representation of C6.452- 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 12 15 91)(2 11 16 96)(3 10 17 95)(4 9 18 94)(5 8 13 93)(6 7 14 92)(19 86 26 79)(20 85 27 84)(21 90 28 83)(22 89 29 82)(23 88 30 81)(24 87 25 80)(31 77 38 70)(32 76 39 69)(33 75 40 68)(34 74 41 67)(35 73 42 72)(36 78 37 71)(43 58 50 65)(44 57 51 64)(45 56 52 63)(46 55 53 62)(47 60 54 61)(48 59 49 66)
(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 50 18 46)(2 51 13 47)(3 52 14 48)(4 53 15 43)(5 54 16 44)(6 49 17 45)(7 63 95 59)(8 64 96 60)(9 65 91 55)(10 66 92 56)(11 61 93 57)(12 62 94 58)(19 31 29 41)(20 32 30 42)(21 33 25 37)(22 34 26 38)(23 35 27 39)(24 36 28 40)(67 89 77 79)(68 90 78 80)(69 85 73 81)(70 86 74 82)(71 87 75 83)(72 88 76 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), (1,12,15,91)(2,11,16,96)(3,10,17,95)(4,9,18,94)(5,8,13,93)(6,7,14,92)(19,86,26,79)(20,85,27,84)(21,90,28,83)(22,89,29,82)(23,88,30,81)(24,87,25,80)(31,77,38,70)(32,76,39,69)(33,75,40,68)(34,74,41,67)(35,73,42,72)(36,78,37,71)(43,58,50,65)(44,57,51,64)(45,56,52,63)(46,55,53,62)(47,60,54,61)(48,59,49,66), (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), (1,50,18,46)(2,51,13,47)(3,52,14,48)(4,53,15,43)(5,54,16,44)(6,49,17,45)(7,63,95,59)(8,64,96,60)(9,65,91,55)(10,66,92,56)(11,61,93,57)(12,62,94,58)(19,31,29,41)(20,32,30,42)(21,33,25,37)(22,34,26,38)(23,35,27,39)(24,36,28,40)(67,89,77,79)(68,90,78,80)(69,85,73,81)(70,86,74,82)(71,87,75,83)(72,88,76,84), (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), (1,12,15,91)(2,11,16,96)(3,10,17,95)(4,9,18,94)(5,8,13,93)(6,7,14,92)(19,86,26,79)(20,85,27,84)(21,90,28,83)(22,89,29,82)(23,88,30,81)(24,87,25,80)(31,77,38,70)(32,76,39,69)(33,75,40,68)(34,74,41,67)(35,73,42,72)(36,78,37,71)(43,58,50,65)(44,57,51,64)(45,56,52,63)(46,55,53,62)(47,60,54,61)(48,59,49,66), (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), (1,50,18,46)(2,51,13,47)(3,52,14,48)(4,53,15,43)(5,54,16,44)(6,49,17,45)(7,63,95,59)(8,64,96,60)(9,65,91,55)(10,66,92,56)(11,61,93,57)(12,62,94,58)(19,31,29,41)(20,32,30,42)(21,33,25,37)(22,34,26,38)(23,35,27,39)(24,36,28,40)(67,89,77,79)(68,90,78,80)(69,85,73,81)(70,86,74,82)(71,87,75,83)(72,88,76,84), (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)], [(1,12,15,91),(2,11,16,96),(3,10,17,95),(4,9,18,94),(5,8,13,93),(6,7,14,92),(19,86,26,79),(20,85,27,84),(21,90,28,83),(22,89,29,82),(23,88,30,81),(24,87,25,80),(31,77,38,70),(32,76,39,69),(33,75,40,68),(34,74,41,67),(35,73,42,72),(36,78,37,71),(43,58,50,65),(44,57,51,64),(45,56,52,63),(46,55,53,62),(47,60,54,61),(48,59,49,66)], [(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)], [(1,50,18,46),(2,51,13,47),(3,52,14,48),(4,53,15,43),(5,54,16,44),(6,49,17,45),(7,63,95,59),(8,64,96,60),(9,65,91,55),(10,66,92,56),(11,61,93,57),(12,62,94,58),(19,31,29,41),(20,32,30,42),(21,33,25,37),(22,34,26,38),(23,35,27,39),(24,36,28,40),(67,89,77,79),(68,90,78,80),(69,85,73,81),(70,86,74,82),(71,87,75,83),(72,88,76,84)], [(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 2G 2H 3 4A ··· 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 6A ··· 6G 12A ··· 12L 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 12 ··· 12 size 1 1 1 1 2 2 12 12 12 2 2 ··· 2 4 4 4 6 6 6 6 12 12 12 2 ··· 2 4 ··· 4

45 irreducible representations

 dim 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 S3 D4 D6 D6 C4○D4 C3⋊D4 2- 1+4 Q8⋊3S3 Q8.15D6 kernel C6.452- 1+4 C4×C3⋊D4 C12⋊7D4 Q8×Dic3 D6⋊3Q8 C12.23D4 C2×Q8⋊3S3 Q8×C2×C6 C22×Q8 C3×Q8 C22×C4 C2×Q8 C2×C6 Q8 C6 C22 C2 # reps 1 3 3 1 3 3 1 1 1 4 3 4 4 8 1 2 2

Matrix representation of C6.452- 1+4 in GL4(𝔽13) generated by

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

C6.452- 1+4 in GAP, Magma, Sage, TeX

`C_6._{45}2_-^{1+4}`
`% in TeX`

`G:=Group("C6.45ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1376);`
`// by ID`

`G=gap.SmallGroup(192,1376);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,387,184,675,6278]);`
`// 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=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`

׿
×
𝔽