Copied to
clipboard

## G = C6.772- 1+4order 192 = 26·3

### 32nd 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 — C6.772- 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — C2×C3⋊D4 — C4×C3⋊D4 — C6.772- 1+4
 Lower central C3 — C2×C6 — C6.772- 1+4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for C6.772- 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=b2, e2=a3b2, bab-1=cac=dad-1=a-1, ae=ea, cbc=b-1, dbd-1=a3b, be=eb, dcd-1=a3c, ce=ec, ede-1=a3b2d >

Subgroups: 464 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, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4×S3, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×Q8, C22×S3, C22×C6, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22⋊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, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C22×C12, C6×Q8, C22.46C24, C23.16D6, C23.21D6, C4.Dic6, C4.Dic6, C4⋊C47S3, C4⋊C4⋊S3, C2×C4⋊Dic3, C4×C3⋊D4, Q8×Dic3, D63Q8, C3×C22⋊Q8, C6.772- 1+4
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2- 1+4, D42S3, Q83S3, S3×C23, C22.46C24, C2×D42S3, C2×Q83S3, Q8○D12, C6.772- 1+4

Smallest permutation representation of C6.772- 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 65 18 55)(2 64 13 60)(3 63 14 59)(4 62 15 58)(5 61 16 57)(6 66 17 56)(7 49 95 45)(8 54 96 44)(9 53 91 43)(10 52 92 48)(11 51 93 47)(12 50 94 46)(19 77 29 67)(20 76 30 72)(21 75 25 71)(22 74 26 70)(23 73 27 69)(24 78 28 68)(31 82 41 86)(32 81 42 85)(33 80 37 90)(34 79 38 89)(35 84 39 88)(36 83 40 87)
(2 6)(3 5)(7 8)(9 12)(10 11)(13 17)(14 16)(19 26)(20 25)(21 30)(22 29)(23 28)(24 27)(32 36)(33 35)(37 39)(40 42)(43 50)(44 49)(45 54)(46 53)(47 52)(48 51)(55 65)(56 64)(57 63)(58 62)(59 61)(60 66)(67 70)(68 69)(71 72)(73 78)(74 77)(75 76)(79 89)(80 88)(81 87)(82 86)(83 85)(84 90)(91 94)(92 93)(95 96)
(1 67 18 77)(2 72 13 76)(3 71 14 75)(4 70 15 74)(5 69 16 73)(6 68 17 78)(7 33 95 37)(8 32 96 42)(9 31 91 41)(10 36 92 40)(11 35 93 39)(12 34 94 38)(19 58 29 62)(20 57 30 61)(21 56 25 66)(22 55 26 65)(23 60 27 64)(24 59 28 63)(43 89 53 79)(44 88 54 84)(45 87 49 83)(46 86 50 82)(47 85 51 81)(48 90 52 80)
(1 31 15 38)(2 32 16 39)(3 33 17 40)(4 34 18 41)(5 35 13 42)(6 36 14 37)(7 75 92 68)(8 76 93 69)(9 77 94 70)(10 78 95 71)(11 73 96 72)(12 74 91 67)(19 50 26 43)(20 51 27 44)(21 52 28 45)(22 53 29 46)(23 54 30 47)(24 49 25 48)(55 86 62 79)(56 87 63 80)(57 88 64 81)(58 89 65 82)(59 90 66 83)(60 85 61 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), (1,65,18,55)(2,64,13,60)(3,63,14,59)(4,62,15,58)(5,61,16,57)(6,66,17,56)(7,49,95,45)(8,54,96,44)(9,53,91,43)(10,52,92,48)(11,51,93,47)(12,50,94,46)(19,77,29,67)(20,76,30,72)(21,75,25,71)(22,74,26,70)(23,73,27,69)(24,78,28,68)(31,82,41,86)(32,81,42,85)(33,80,37,90)(34,79,38,89)(35,84,39,88)(36,83,40,87), (2,6)(3,5)(7,8)(9,12)(10,11)(13,17)(14,16)(19,26)(20,25)(21,30)(22,29)(23,28)(24,27)(32,36)(33,35)(37,39)(40,42)(43,50)(44,49)(45,54)(46,53)(47,52)(48,51)(55,65)(56,64)(57,63)(58,62)(59,61)(60,66)(67,70)(68,69)(71,72)(73,78)(74,77)(75,76)(79,89)(80,88)(81,87)(82,86)(83,85)(84,90)(91,94)(92,93)(95,96), (1,67,18,77)(2,72,13,76)(3,71,14,75)(4,70,15,74)(5,69,16,73)(6,68,17,78)(7,33,95,37)(8,32,96,42)(9,31,91,41)(10,36,92,40)(11,35,93,39)(12,34,94,38)(19,58,29,62)(20,57,30,61)(21,56,25,66)(22,55,26,65)(23,60,27,64)(24,59,28,63)(43,89,53,79)(44,88,54,84)(45,87,49,83)(46,86,50,82)(47,85,51,81)(48,90,52,80), (1,31,15,38)(2,32,16,39)(3,33,17,40)(4,34,18,41)(5,35,13,42)(6,36,14,37)(7,75,92,68)(8,76,93,69)(9,77,94,70)(10,78,95,71)(11,73,96,72)(12,74,91,67)(19,50,26,43)(20,51,27,44)(21,52,28,45)(22,53,29,46)(23,54,30,47)(24,49,25,48)(55,86,62,79)(56,87,63,80)(57,88,64,81)(58,89,65,82)(59,90,66,83)(60,85,61,84)>;`

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + - - + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 D6 C4○D4 C4○D4 2- 1+4 D4⋊2S3 Q8⋊3S3 Q8○D12 kernel C6.772- 1+4 C23.16D6 C23.21D6 C4.Dic6 C4⋊C4⋊7S3 C4⋊C4⋊S3 C2×C4⋊Dic3 C4×C3⋊D4 Q8×Dic3 D6⋊3Q8 C3×C22⋊Q8 C22⋊Q8 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C12 C2×C6 C6 C4 C22 C2 # reps 1 2 2 3 1 2 1 1 1 1 1 1 2 3 1 1 4 4 1 2 2 2

Matrix representation of C6.772- 1+4 in GL6(𝔽13)

 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 0 12 0 0 0 0 1 12
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 12 0
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 8 0 0 0 0 8 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 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 8 0 0 0 0 0 0 5 0 0 0 0 0 0 1 0 0 0 0 0 0 1

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

C6.772- 1+4 in GAP, Magma, Sage, TeX

`C_6._{77}2_-^{1+4}`
`% in TeX`

`G:=Group("C6.77ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1201);`
`// by ID`

`G=gap.SmallGroup(192,1201);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,100,675,570,185,192,6278]);`
`// 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=c*a*c=d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,d*b*d^-1=a^3*b,b*e=e*b,d*c*d^-1=a^3*c,c*e=e*c,e*d*e^-1=a^3*b^2*d>;`
`// generators/relations`

׿
×
𝔽