Copied to
clipboard

G = C6.592+ 1+4order 192 = 26·3

59th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C6.592+ 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — D6⋊3Q8 — C6.592+ 1+4
 Lower central C3 — C2×C6 — C6.592+ 1+4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for C6.592+ 1+4
G = < a,b,c,d,e | a6=b4=c2=e2=1, d2=a3b2, ab=ba, ac=ca, dad-1=eae=a-1, cbc=a3b-1, dbd-1=a3b, be=eb, dcd-1=ece=a3c, ede=b2d >

Subgroups: 624 in 220 conjugacy classes, 91 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×Q8, C22×S3, C22×C6, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C2×C3⋊D4, C22×C12, C6×Q8, C22.56C24, C23.9D6, Dic3⋊D4, C23.11D6, Dic3.Q8, D6.D4, C12⋊D4, D6⋊Q8, C4.D12, C23.28D6, C127D4, D63Q8, C12.23D4, C3×C22⋊Q8, C6.592+ 1+4
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, 2- 1+4, S3×C23, C22.56C24, D46D6, Q8.15D6, D4○D12, C6.592+ 1+4

Smallest permutation representation of C6.592+ 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 93 14 9)(2 94 15 10)(3 95 16 11)(4 96 17 12)(5 91 18 7)(6 92 13 8)(19 82 27 90)(20 83 28 85)(21 84 29 86)(22 79 30 87)(23 80 25 88)(24 81 26 89)(31 67 39 75)(32 68 40 76)(33 69 41 77)(34 70 42 78)(35 71 37 73)(36 72 38 74)(43 55 51 63)(44 56 52 64)(45 57 53 65)(46 58 54 66)(47 59 49 61)(48 60 50 62)
(1 24)(2 19)(3 20)(4 21)(5 22)(6 23)(7 82)(8 83)(9 84)(10 79)(11 80)(12 81)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 51)(40 52)(41 53)(42 54)(55 78)(56 73)(57 74)(58 75)(59 76)(60 77)(61 68)(62 69)(63 70)(64 71)(65 72)(66 67)(85 92)(86 93)(87 94)(88 95)(89 96)(90 91)
(1 93 17 12)(2 92 18 11)(3 91 13 10)(4 96 14 9)(5 95 15 8)(6 94 16 7)(19 88 30 83)(20 87 25 82)(21 86 26 81)(22 85 27 80)(23 90 28 79)(24 89 29 84)(31 76 42 71)(32 75 37 70)(33 74 38 69)(34 73 39 68)(35 78 40 67)(36 77 41 72)(43 56 54 61)(44 55 49 66)(45 60 50 65)(46 59 51 64)(47 58 52 63)(48 57 53 62)
(1 69)(2 68)(3 67)(4 72)(5 71)(6 70)(7 35)(8 34)(9 33)(10 32)(11 31)(12 36)(13 78)(14 77)(15 76)(16 75)(17 74)(18 73)(19 64)(20 63)(21 62)(22 61)(23 66)(24 65)(25 58)(26 57)(27 56)(28 55)(29 60)(30 59)(37 91)(38 96)(39 95)(40 94)(41 93)(42 92)(43 83)(44 82)(45 81)(46 80)(47 79)(48 84)(49 87)(50 86)(51 85)(52 90)(53 89)(54 88)```

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

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

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

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 D6 2+ 1+4 2- 1+4 D4⋊6D6 Q8.15D6 D4○D12 kernel C6.592+ 1+4 C23.9D6 Dic3⋊D4 C23.11D6 Dic3.Q8 D6.D4 C12⋊D4 D6⋊Q8 C4.D12 C23.28D6 C12⋊7D4 D6⋊3Q8 C12.23D4 C3×C22⋊Q8 C22⋊Q8 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C6 C6 C2 C2 C2 # reps 1 1 2 1 1 2 1 1 1 1 1 1 1 1 1 2 3 1 1 2 1 2 2 2

Matrix representation of C6.592+ 1+4 in GL8(𝔽13)

 1 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 12 0
,
 0 0 3 6 0 0 0 0 0 0 7 10 0 0 0 0 3 6 0 0 0 0 0 0 7 10 0 0 0 0 0 0 0 0 0 0 0 10 6 12 0 0 0 0 3 3 1 7 0 0 0 0 12 3 0 3 0 0 0 0 10 9 10 10
,
 11 9 0 0 0 0 0 0 4 2 0 0 0 0 0 0 0 0 11 9 0 0 0 0 0 0 4 2 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 5 9 12 0 0 0 0 0 4 9 0 12
,
 0 0 10 7 0 0 0 0 0 0 10 3 0 0 0 0 10 7 0 0 0 0 0 0 10 3 0 0 0 0 0 0 0 0 0 0 0 3 7 1 0 0 0 0 3 0 7 6 0 0 0 0 2 4 10 10 0 0 0 0 2 11 0 3
,
 3 6 0 0 0 0 0 0 3 10 0 0 0 0 0 0 0 0 10 7 0 0 0 0 0 0 10 3 0 0 0 0 0 0 0 0 8 4 2 0 0 0 0 0 9 5 11 11 0 0 0 0 9 0 9 4 0 0 0 0 4 4 8 4

`G:=sub<GL(8,GF(13))| [1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0],[0,0,3,7,0,0,0,0,0,0,6,10,0,0,0,0,3,7,0,0,0,0,0,0,6,10,0,0,0,0,0,0,0,0,0,0,0,3,12,10,0,0,0,0,10,3,3,9,0,0,0,0,6,1,0,10,0,0,0,0,12,7,3,10],[11,4,0,0,0,0,0,0,9,2,0,0,0,0,0,0,0,0,11,4,0,0,0,0,0,0,9,2,0,0,0,0,0,0,0,0,1,0,5,4,0,0,0,0,0,1,9,9,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[0,0,10,10,0,0,0,0,0,0,7,3,0,0,0,0,10,10,0,0,0,0,0,0,7,3,0,0,0,0,0,0,0,0,0,0,0,3,2,2,0,0,0,0,3,0,4,11,0,0,0,0,7,7,10,0,0,0,0,0,1,6,10,3],[3,3,0,0,0,0,0,0,6,10,0,0,0,0,0,0,0,0,10,10,0,0,0,0,0,0,7,3,0,0,0,0,0,0,0,0,8,9,9,4,0,0,0,0,4,5,0,4,0,0,0,0,2,11,9,8,0,0,0,0,0,11,4,4] >;`

C6.592+ 1+4 in GAP, Magma, Sage, TeX

`C_6._{59}2_+^{1+4}`
`% in TeX`

`G:=Group("C6.59ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1206);`
`// by ID`

`G=gap.SmallGroup(192,1206);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,219,100,1571,570,6278]);`
`// Polycyclic`

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

׿
×
𝔽