Copied to
clipboard

## G = C6.702- 1+4order 192 = 26·3

### 25th 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 — C6.702- 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — D4×Dic3 — C6.702- 1+4
 Lower central C3 — C2×C6 — C6.702- 1+4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C6.702- 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=a3b2, e2=b2, ab=ba, ac=ca, dad-1=eae-1=a-1, cbc=b-1, bd=db, ebe-1=a3b, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 496 in 218 conjugacy classes, 95 normal (31 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C422C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C22×Dic3, C22×Dic3, C22×C12, C6×D4, C6×D4, C22.33C24, Dic3.D4, C23.8D6, Dic3.Q8, C2×Dic3⋊C4, C12.48D4, D4×Dic3, C23.23D6, C3×C4⋊D4, C6.702- 1+4
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, 2- 1+4, D42S3, S3×C23, C22.33C24, C2×D42S3, D46D6, Q8○D12, C6.702- 1+4

Smallest permutation representation of C6.702- 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 51 13 43)(2 52 14 44)(3 53 15 45)(4 54 16 46)(5 49 17 47)(6 50 18 48)(7 64 92 56)(8 65 93 57)(9 66 94 58)(10 61 95 59)(11 62 96 60)(12 63 91 55)(19 31 26 38)(20 32 27 39)(21 33 28 40)(22 34 29 41)(23 35 30 42)(24 36 25 37)(67 87 75 79)(68 88 76 80)(69 89 77 81)(70 90 78 82)(71 85 73 83)(72 86 74 84)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 37)(7 76)(8 77)(9 78)(10 73)(11 74)(12 75)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 50)(26 51)(27 52)(28 53)(29 54)(30 49)(55 79)(56 80)(57 81)(58 82)(59 83)(60 84)(61 85)(62 86)(63 87)(64 88)(65 89)(66 90)(67 91)(68 92)(69 93)(70 94)(71 95)(72 96)
(1 80 16 85)(2 79 17 90)(3 84 18 89)(4 83 13 88)(5 82 14 87)(6 81 15 86)(7 29 95 19)(8 28 96 24)(9 27 91 23)(10 26 92 22)(11 25 93 21)(12 30 94 20)(31 64 41 59)(32 63 42 58)(33 62 37 57)(34 61 38 56)(35 66 39 55)(36 65 40 60)(43 76 54 71)(44 75 49 70)(45 74 50 69)(46 73 51 68)(47 78 52 67)(48 77 53 72)
(1 73 13 71)(2 78 14 70)(3 77 15 69)(4 76 16 68)(5 75 17 67)(6 74 18 72)(7 34 92 41)(8 33 93 40)(9 32 94 39)(10 31 95 38)(11 36 96 37)(12 35 91 42)(19 64 26 56)(20 63 27 55)(21 62 28 60)(22 61 29 59)(23 66 30 58)(24 65 25 57)(43 88 51 80)(44 87 52 79)(45 86 53 84)(46 85 54 83)(47 90 49 82)(48 89 50 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), (1,51,13,43)(2,52,14,44)(3,53,15,45)(4,54,16,46)(5,49,17,47)(6,50,18,48)(7,64,92,56)(8,65,93,57)(9,66,94,58)(10,61,95,59)(11,62,96,60)(12,63,91,55)(19,31,26,38)(20,32,27,39)(21,33,28,40)(22,34,29,41)(23,35,30,42)(24,36,25,37)(67,87,75,79)(68,88,76,80)(69,89,77,81)(70,90,78,82)(71,85,73,83)(72,86,74,84), (1,38)(2,39)(3,40)(4,41)(5,42)(6,37)(7,76)(8,77)(9,78)(10,73)(11,74)(12,75)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,50)(26,51)(27,52)(28,53)(29,54)(30,49)(55,79)(56,80)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96), (1,80,16,85)(2,79,17,90)(3,84,18,89)(4,83,13,88)(5,82,14,87)(6,81,15,86)(7,29,95,19)(8,28,96,24)(9,27,91,23)(10,26,92,22)(11,25,93,21)(12,30,94,20)(31,64,41,59)(32,63,42,58)(33,62,37,57)(34,61,38,56)(35,66,39,55)(36,65,40,60)(43,76,54,71)(44,75,49,70)(45,74,50,69)(46,73,51,68)(47,78,52,67)(48,77,53,72), (1,73,13,71)(2,78,14,70)(3,77,15,69)(4,76,16,68)(5,75,17,67)(6,74,18,72)(7,34,92,41)(8,33,93,40)(9,32,94,39)(10,31,95,38)(11,36,96,37)(12,35,91,42)(19,64,26,56)(20,63,27,55)(21,62,28,60)(22,61,29,59)(23,66,30,58)(24,65,25,57)(43,88,51,80)(44,87,52,79)(45,86,53,84)(46,85,54,83)(47,90,49,82)(48,89,50,81)>;`

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4N 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D 12E 12F order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 ··· 4 6 6 6 6 6 6 6 12 12 12 12 12 12 size 1 1 1 1 2 2 4 4 2 4 4 4 4 6 6 6 6 12 ··· 12 2 2 2 4 4 8 8 4 4 4 4 8 8

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + - - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 D6 C4○D4 2+ 1+4 2- 1+4 D4⋊2S3 D4⋊6D6 Q8○D12 kernel C6.702- 1+4 Dic3.D4 C23.8D6 Dic3.Q8 C2×Dic3⋊C4 C12.48D4 D4×Dic3 C23.23D6 C3×C4⋊D4 C4⋊D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C2×C6 C6 C6 C22 C2 C2 # reps 1 2 2 2 1 1 2 4 1 1 2 1 1 3 4 1 1 2 2 2

Matrix representation of C6.702- 1+4 in GL8(𝔽13)

 0 1 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 0 1 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 0 0 0 0 0 0 1 0 0 10 0 0 0 0 0 0 12 12 0 0 0 0 8 1 0 1 0 0 0 0 5 0 0 12
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 3 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 1 0
,
 9 2 0 0 0 0 0 0 11 4 0 0 0 0 0 0 0 0 9 2 0 0 0 0 0 0 11 4 0 0 0 0 0 0 0 0 8 0 11 0 0 0 0 0 1 0 8 5 0 0 0 0 0 0 5 0 0 0 0 0 1 5 8 0
,
 0 0 9 2 0 0 0 0 0 0 11 4 0 0 0 0 4 11 0 0 0 0 0 0 2 9 0 0 0 0 0 0 0 0 0 0 5 11 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 5 0 8 0 0 0 0 0 8 8 0

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

C6.702- 1+4 in GAP, Magma, Sage, TeX

`C_6._{70}2_-^{1+4}`
`% in TeX`

`G:=Group("C6.70ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1161);`
`// by ID`

`G=gap.SmallGroup(192,1161);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,219,675,297,136,6278]);`
`// 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=e*a*e^-1=a^-1,c*b*c=b^-1,b*d=d*b,e*b*e^-1=a^3*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;`
`// generators/relations`

׿
×
𝔽