Copied to
clipboard

## G = C6.322+ 1+4order 192 = 26·3

### 32nd 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 — C6.322+ 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C2×D4⋊2S3 — C6.322+ 1+4
 Lower central C3 — C2×C6 — C6.322+ 1+4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C6.322+ 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=e2=a3b2, bab-1=dad-1=a-1, ac=ca, ae=ea, cbc=b-1, dbd-1=ebe-1=a3b, cd=dc, ece-1=a3c, ede-1=b2d >

Subgroups: 784 in 294 conjugacy classes, 103 normal (31 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C2×C4⋊C4, C4⋊D4, C4⋊D4, C22⋊Q8, C2×C4○D4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, D42S3, C22×Dic3, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, C22.31C24, Dic3.D4, Dic3⋊D4, D6⋊Q8, C2×Dic3⋊C4, C127D4, C23.14D6, C3×C4⋊D4, C2×D42S3, C6.322+ 1+4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, 2+ 1+4, 2- 1+4, S3×D4, S3×C23, C22.31C24, C2×S3×D4, D46D6, Q8○D12, C6.322+ 1+4

Smallest permutation representation of C6.322+ 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 26 15 19)(2 25 16 24)(3 30 17 23)(4 29 18 22)(5 28 13 21)(6 27 14 20)(7 81 92 88)(8 80 93 87)(9 79 94 86)(10 84 95 85)(11 83 96 90)(12 82 91 89)(31 46 38 53)(32 45 39 52)(33 44 40 51)(34 43 41 50)(35 48 42 49)(36 47 37 54)(55 67 62 74)(56 72 63 73)(57 71 64 78)(58 70 65 77)(59 69 66 76)(60 68 61 75)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 25)(7 90)(8 85)(9 86)(10 87)(11 88)(12 89)(13 23)(14 24)(15 19)(16 20)(17 21)(18 22)(31 50)(32 51)(33 52)(34 53)(35 54)(36 49)(37 48)(38 43)(39 44)(40 45)(41 46)(42 47)(55 77)(56 78)(57 73)(58 74)(59 75)(60 76)(61 69)(62 70)(63 71)(64 72)(65 67)(66 68)(79 94)(80 95)(81 96)(82 91)(83 92)(84 93)
(1 43 18 53)(2 48 13 52)(3 47 14 51)(4 46 15 50)(5 45 16 49)(6 44 17 54)(7 60 95 64)(8 59 96 63)(9 58 91 62)(10 57 92 61)(11 56 93 66)(12 55 94 65)(19 31 29 41)(20 36 30 40)(21 35 25 39)(22 34 26 38)(23 33 27 37)(24 32 28 42)(67 89 77 79)(68 88 78 84)(69 87 73 83)(70 86 74 82)(71 85 75 81)(72 90 76 80)
(1 86 18 82)(2 87 13 83)(3 88 14 84)(4 89 15 79)(5 90 16 80)(6 85 17 81)(7 23 95 27)(8 24 96 28)(9 19 91 29)(10 20 92 30)(11 21 93 25)(12 22 94 26)(31 55 41 65)(32 56 42 66)(33 57 37 61)(34 58 38 62)(35 59 39 63)(36 60 40 64)(43 67 53 77)(44 68 54 78)(45 69 49 73)(46 70 50 74)(47 71 51 75)(48 72 52 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), (1,26,15,19)(2,25,16,24)(3,30,17,23)(4,29,18,22)(5,28,13,21)(6,27,14,20)(7,81,92,88)(8,80,93,87)(9,79,94,86)(10,84,95,85)(11,83,96,90)(12,82,91,89)(31,46,38,53)(32,45,39,52)(33,44,40,51)(34,43,41,50)(35,48,42,49)(36,47,37,54)(55,67,62,74)(56,72,63,73)(57,71,64,78)(58,70,65,77)(59,69,66,76)(60,68,61,75), (1,26)(2,27)(3,28)(4,29)(5,30)(6,25)(7,90)(8,85)(9,86)(10,87)(11,88)(12,89)(13,23)(14,24)(15,19)(16,20)(17,21)(18,22)(31,50)(32,51)(33,52)(34,53)(35,54)(36,49)(37,48)(38,43)(39,44)(40,45)(41,46)(42,47)(55,77)(56,78)(57,73)(58,74)(59,75)(60,76)(61,69)(62,70)(63,71)(64,72)(65,67)(66,68)(79,94)(80,95)(81,96)(82,91)(83,92)(84,93), (1,43,18,53)(2,48,13,52)(3,47,14,51)(4,46,15,50)(5,45,16,49)(6,44,17,54)(7,60,95,64)(8,59,96,63)(9,58,91,62)(10,57,92,61)(11,56,93,66)(12,55,94,65)(19,31,29,41)(20,36,30,40)(21,35,25,39)(22,34,26,38)(23,33,27,37)(24,32,28,42)(67,89,77,79)(68,88,78,84)(69,87,73,83)(70,86,74,82)(71,85,75,81)(72,90,76,80), (1,86,18,82)(2,87,13,83)(3,88,14,84)(4,89,15,79)(5,90,16,80)(6,85,17,81)(7,23,95,27)(8,24,96,28)(9,19,91,29)(10,20,92,30)(11,21,93,25)(12,22,94,26)(31,55,41,65)(32,56,42,66)(33,57,37,61)(34,58,38,62)(35,59,39,63)(36,60,40,64)(43,67,53,77)(44,68,54,78)(45,69,49,73)(46,70,50,74)(47,71,51,75)(48,72,52,76)>;`

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D 12E 12F order 1 2 2 2 2 2 2 2 2 2 3 4 4 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 12 12 2 4 4 4 4 6 6 6 6 12 12 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 D4 D6 D6 D6 D6 2+ 1+4 2- 1+4 S3×D4 D4⋊6D6 Q8○D12 kernel C6.322+ 1+4 Dic3.D4 Dic3⋊D4 D6⋊Q8 C2×Dic3⋊C4 C12⋊7D4 C23.14D6 C3×C4⋊D4 C2×D4⋊2S3 C4⋊D4 C2×Dic3 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C6 C6 C22 C2 C2 # reps 1 2 2 2 1 1 4 1 2 1 4 2 1 1 3 1 1 2 2 2

Matrix representation of C6.322+ 1+4 in GL8(𝔽13)

 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12
,
 1 10 0 0 0 0 0 0 5 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0
,
 12 3 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 1 0 0 0 0 1 0 1 0
,
 1 10 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 12 0 2 0 0 0 0 1 0 2 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0
,
 12 0 0 0 0 0 0 0 8 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 3 6 0 0 0 0 0 0 7 10 0 0 0 0 0 0 10 0 10 7 0 0 0 0 0 10 6 3

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

C6.322+ 1+4 in GAP, Magma, Sage, TeX

`C_6._{32}2_+^{1+4}`
`% in TeX`

`G:=Group("C6.32ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1156);`
`// by ID`

`G=gap.SmallGroup(192,1156);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,387,1123,570,185,6278]);`
`// Polycyclic`

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

׿
×
𝔽