Copied to
clipboard

## G = C6.1052- 1+4order 192 = 26·3

### 60th 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 — C6.1052- 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C22×Dic6 — C6.1052- 1+4
 Lower central C3 — C2×C6 — C6.1052- 1+4
 Upper central C1 — C22 — C2×C4○D4

Generators and relations for C6.1052- 1+4
G = < a,b,c,d,e | a6=b4=1, c2=a3, d2=b2, e2=a3b2, bab-1=cac-1=eae-1=a-1, ad=da, cbc-1=a3b-1, dbd-1=ebe-1=a3b, dcd-1=a3c, ce=ec, ede-1=a3b2d >

Subgroups: 584 in 270 conjugacy classes, 111 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×4], C4 [×10], C22, C22 [×2], C22 [×8], C6, C6 [×2], C6 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×16], D4 [×6], Q8 [×10], C23, C23 [×2], Dic3 [×8], C12 [×4], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×8], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×10], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×8], C4○D4 [×4], Dic6 [×8], C2×Dic3 [×8], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×6], C2×C12 [×4], C3×D4 [×6], C3×Q8 [×2], C22×C6, C22×C6 [×2], C42⋊C2, C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×2], C4⋊Q8 [×2], C22×Q8, C2×C4○D4, C4×Dic3 [×2], Dic3⋊C4 [×8], C4⋊Dic3 [×2], C6.D4 [×10], C2×Dic6 [×4], C2×Dic6 [×4], C22×Dic3 [×2], C22×C12, C22×C12 [×2], C6×D4, C6×D4 [×2], C6×Q8, C3×C4○D4 [×4], C23.38C23, C12.48D4 [×4], C23.26D6, C23.23D6 [×4], C23.12D6 [×2], Dic3⋊Q8 [×2], C22×Dic6, C6×C4○D4, C6.1052- 1+4
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C3⋊D4 [×4], C22×S3 [×7], C22×D4, 2- 1+4 [×2], C2×C3⋊D4 [×6], S3×C23, C23.38C23, Q8○D12 [×2], C22×C3⋊D4, C6.1052- 1+4

Smallest permutation representation of C6.1052- 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 76 17 72)(2 75 18 71)(3 74 13 70)(4 73 14 69)(5 78 15 68)(6 77 16 67)(7 41 96 36)(8 40 91 35)(9 39 92 34)(10 38 93 33)(11 37 94 32)(12 42 95 31)(19 59 30 63)(20 58 25 62)(21 57 26 61)(22 56 27 66)(23 55 28 65)(24 60 29 64)(43 83 49 87)(44 82 50 86)(45 81 51 85)(46 80 52 90)(47 79 53 89)(48 84 54 88)
(1 84 4 81)(2 83 5 80)(3 82 6 79)(7 26 10 29)(8 25 11 28)(9 30 12 27)(13 86 16 89)(14 85 17 88)(15 90 18 87)(19 95 22 92)(20 94 23 91)(21 93 24 96)(31 63 34 66)(32 62 35 65)(33 61 36 64)(37 58 40 55)(38 57 41 60)(39 56 42 59)(43 75 46 78)(44 74 47 77)(45 73 48 76)(49 71 52 68)(50 70 53 67)(51 69 54 72)
(1 33 17 38)(2 34 18 39)(3 35 13 40)(4 36 14 41)(5 31 15 42)(6 32 16 37)(7 72 96 76)(8 67 91 77)(9 68 92 78)(10 69 93 73)(11 70 94 74)(12 71 95 75)(19 46 30 52)(20 47 25 53)(21 48 26 54)(22 43 27 49)(23 44 28 50)(24 45 29 51)(55 79 65 89)(56 80 66 90)(57 81 61 85)(58 82 62 86)(59 83 63 87)(60 84 64 88)
(1 76 14 69)(2 75 15 68)(3 74 16 67)(4 73 17 72)(5 78 18 71)(6 77 13 70)(7 33 93 41)(8 32 94 40)(9 31 95 39)(10 36 96 38)(11 35 91 37)(12 34 92 42)(19 59 27 66)(20 58 28 65)(21 57 29 64)(22 56 30 63)(23 55 25 62)(24 60 26 61)(43 87 52 80)(44 86 53 79)(45 85 54 84)(46 90 49 83)(47 89 50 82)(48 88 51 81)```

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 C3⋊D4 2- 1+4 Q8○D12 kernel C6.1052- 1+4 C12.48D4 C23.26D6 C23.23D6 C23.12D6 Dic3⋊Q8 C22×Dic6 C6×C4○D4 C2×C4○D4 C2×C12 C22×C4 C2×D4 C2×Q8 C2×C4 C6 C2 # reps 1 4 1 4 2 2 1 1 1 4 3 3 1 8 2 4

Matrix representation of C6.1052- 1+4 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 3 0 0 0 0 0 0 3 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 8 0 0 0 0 0 0 8 0 0
,
 0 12 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 4 0 0 0 0 6 1 0 0 1 9 0 0 0 0 7 12 0 0
,
 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 9 0 0 0 0 7 12 0 0 0 0 0 0 1 9 0 0 0 0 7 12
,
 0 12 0 0 0 0 1 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 4 5 0 0 8 0 0 0 0 0 4 5 0 0

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

C6.1052- 1+4 in GAP, Magma, Sage, TeX

`C_6._{105}2_-^{1+4}`
`% in TeX`

`G:=Group("C6.105ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1384);`
`// by ID`

`G=gap.SmallGroup(192,1384);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,758,184,675,570,6278]);`
`// Polycyclic`

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

׿
×
𝔽