Copied to
clipboard

## G = C6.442- 1+4order 192 = 26·3

### 44th 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 — C6.442- 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C2×C4○D12 — C6.442- 1+4
 Lower central C3 — C2×C6 — C6.442- 1+4
 Upper central C1 — C22 — C22×Q8

Generators and relations for C6.442- 1+4
G = < a,b,c,d,e | a6=b4=1, c2=a3, d2=e2=a3b2, bab-1=cac-1=a-1, ad=da, ae=ea, cbc-1=a3b-1, dbd-1=ebe-1=a3b, dcd-1=ece-1=a3c, ede-1=a3b2d >

Subgroups: 616 in 270 conjugacy classes, 111 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×4], C4 [×10], C22, C22 [×2], C22 [×8], S3 [×2], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×14], D4 [×6], Q8 [×10], C23, C23 [×2], Dic3 [×6], C12 [×4], C12 [×4], D6 [×6], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×10], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×3], C2×Q8 [×4], C2×Q8 [×5], C4○D4 [×4], Dic6 [×2], C4×S3 [×4], D12 [×2], C2×Dic3 [×6], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×8], C2×C12 [×4], C3×Q8 [×8], C22×S3 [×2], C22×C6, 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], D6⋊C4 [×8], C6.D4 [×2], C2×Dic6, S3×C2×C4 [×2], C2×D12, C4○D12 [×4], C2×C3⋊D4 [×2], C22×C12, C22×C12 [×2], C6×Q8 [×4], C6×Q8 [×4], C23.38C23, C23.26D6, C23.28D6 [×4], Dic3⋊Q8 [×2], D63Q8 [×4], C12.23D4 [×2], C2×C4○D12, Q8×C2×C6, C6.442- 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, Q8.15D6 [×2], C22×C3⋊D4, C6.442- 1+4

Smallest permutation representation of C6.442- 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 19 14 29)(2 24 15 28)(3 23 16 27)(4 22 17 26)(5 21 18 25)(6 20 13 30)(7 84 95 88)(8 83 96 87)(9 82 91 86)(10 81 92 85)(11 80 93 90)(12 79 94 89)(31 53 41 43)(32 52 42 48)(33 51 37 47)(34 50 38 46)(35 49 39 45)(36 54 40 44)(55 77 65 67)(56 76 66 72)(57 75 61 71)(58 74 62 70)(59 73 63 69)(60 78 64 68)
(1 43 4 46)(2 48 5 45)(3 47 6 44)(7 60 10 57)(8 59 11 56)(9 58 12 55)(13 54 16 51)(14 53 17 50)(15 52 18 49)(19 38 22 41)(20 37 23 40)(21 42 24 39)(25 32 28 35)(26 31 29 34)(27 36 30 33)(61 95 64 92)(62 94 65 91)(63 93 66 96)(67 79 70 82)(68 84 71 81)(69 83 72 80)(73 87 76 90)(74 86 77 89)(75 85 78 88)
(1 91 17 12)(2 92 18 7)(3 93 13 8)(4 94 14 9)(5 95 15 10)(6 96 16 11)(19 89 26 82)(20 90 27 83)(21 85 28 84)(22 86 29 79)(23 87 30 80)(24 88 25 81)(31 70 38 77)(32 71 39 78)(33 72 40 73)(34 67 41 74)(35 68 42 75)(36 69 37 76)(43 65 50 58)(44 66 51 59)(45 61 52 60)(46 62 53 55)(47 63 54 56)(48 64 49 57)
(1 26 17 19)(2 27 18 20)(3 28 13 21)(4 29 14 22)(5 30 15 23)(6 25 16 24)(7 83 92 90)(8 84 93 85)(9 79 94 86)(10 80 95 87)(11 81 96 88)(12 82 91 89)(31 53 38 46)(32 54 39 47)(33 49 40 48)(34 50 41 43)(35 51 42 44)(36 52 37 45)(55 70 62 77)(56 71 63 78)(57 72 64 73)(58 67 65 74)(59 68 66 75)(60 69 61 76)```

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 C3⋊D4 2- 1+4 Q8.15D6 kernel C6.442- 1+4 C23.26D6 C23.28D6 Dic3⋊Q8 D6⋊3Q8 C12.23D4 C2×C4○D12 Q8×C2×C6 C22×Q8 C2×C12 C22×C4 C2×Q8 C2×C4 C6 C2 # reps 1 1 4 2 4 2 1 1 1 4 3 4 8 2 4

Matrix representation of C6.442- 1+4 in GL6(𝔽13)

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

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

C6.442- 1+4 in GAP, Magma, Sage, TeX

`C_6._{44}2_-^{1+4}`
`% in TeX`

`G:=Group("C6.44ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1375);`
`// by ID`

`G=gap.SmallGroup(192,1375);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,675,570,136,6278]);`
`// Polycyclic`

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

׿
×
𝔽