Copied to
clipboard

## G = C62.29C23order 288 = 25·32

### 24th non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.29C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — D6⋊Dic3 — C62.29C23
 Lower central C32 — C62 — C62.29C23
 Upper central C1 — C22 — C2×C4

Generators and relations for C62.29C23
G = < a,b,c,d,e | a6=b6=c2=1, d2=b3, e2=a3b3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=a3c, ece-1=b3c, de=ed >

Subgroups: 458 in 133 conjugacy classes, 44 normal (all characteristic)
C1, C2 [×3], C2, C3 [×2], C3, C4 [×6], C22, C22 [×3], S3, C6 [×6], C6 [×4], C2×C4, C2×C4 [×5], C23, C32, Dic3 [×10], C12 [×6], D6 [×3], C2×C6 [×2], C2×C6 [×4], C42, C22⋊C4 [×3], C4⋊C4 [×3], C3×S3, C3×C6 [×3], C2×Dic3 [×3], C2×Dic3 [×6], C2×C12 [×2], C2×C12 [×4], C22×S3, C22×C6, C422C2, C3×Dic3 [×3], C3⋊Dic3 [×2], C3×C12, S3×C6 [×3], C62, C4×Dic3, Dic3⋊C4 [×6], C4⋊Dic3, D6⋊C4, D6⋊C4 [×2], C6.D4 [×2], C4×C12, C3×C22⋊C4, C6×Dic3 [×3], C2×C3⋊Dic3 [×2], C6×C12, S3×C2×C6, C423S3, C23.8D6, D6⋊Dic3 [×2], Dic3⋊Dic3, C62.C22, Dic3×C12, C3×D6⋊C4, C6.Dic6, C62.29C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], C23, D6 [×6], C4○D4 [×3], C22×S3 [×2], C422C2, S32, C4○D12 [×4], D42S3 [×2], C2×S32, C423S3, C23.8D6, D125S3, D6.D6, D6.3D6, C62.29C23

Smallest permutation representation of C62.29C23
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 18 5 16 3 14)(2 13 6 17 4 15)(7 95 11 93 9 91)(8 96 12 94 10 92)(19 30 23 28 21 26)(20 25 24 29 22 27)(31 40 33 42 35 38)(32 41 34 37 36 39)(43 53 45 49 47 51)(44 54 46 50 48 52)(55 63 57 65 59 61)(56 64 58 66 60 62)(67 74 69 76 71 78)(68 75 70 77 72 73)(79 86 83 90 81 88)(80 87 84 85 82 89)
(1 60)(2 55)(3 56)(4 57)(5 58)(6 59)(7 51)(8 52)(9 53)(10 54)(11 49)(12 50)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)
(1 41 16 36)(2 40 17 35)(3 39 18 34)(4 38 13 33)(5 37 14 32)(6 42 15 31)(7 75 93 72)(8 74 94 71)(9 73 95 70)(10 78 96 69)(11 77 91 68)(12 76 92 67)(19 53 28 47)(20 52 29 46)(21 51 30 45)(22 50 25 44)(23 49 26 43)(24 54 27 48)(55 85 65 80)(56 90 66 79)(57 89 61 84)(58 88 62 83)(59 87 63 82)(60 86 64 81)
(1 30 13 24)(2 25 14 19)(3 26 15 20)(4 27 16 21)(5 28 17 22)(6 29 18 23)(7 81 96 89)(8 82 91 90)(9 83 92 85)(10 84 93 86)(11 79 94 87)(12 80 95 88)(31 52 39 43)(32 53 40 44)(33 54 41 45)(34 49 42 46)(35 50 37 47)(36 51 38 48)(55 70 62 76)(56 71 63 77)(57 72 64 78)(58 67 65 73)(59 68 66 74)(60 69 61 75)```

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A ··· 6F 6G 6H 6I 6J 6K 12A 12B 12C 12D 12E ··· 12J 12K ··· 12R 12S 12T order 1 2 2 2 2 3 3 3 4 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 6 6 12 12 12 12 12 ··· 12 12 ··· 12 12 12 size 1 1 1 1 12 2 2 4 2 2 6 6 6 6 12 36 36 2 ··· 2 4 4 4 12 12 2 2 2 2 4 ··· 4 6 ··· 6 12 12

48 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 S3 S3 D6 D6 D6 C4○D4 C4○D12 S32 D4⋊2S3 C2×S32 D12⋊5S3 D6.D6 D6.3D6 kernel C62.29C23 D6⋊Dic3 Dic3⋊Dic3 C62.C22 Dic3×C12 C3×D6⋊C4 C6.Dic6 C4×Dic3 D6⋊C4 C2×Dic3 C2×C12 C22×S3 C3×C6 C6 C2×C4 C6 C22 C2 C2 C2 # reps 1 2 1 1 1 1 1 1 1 3 2 1 6 16 1 2 1 2 2 2

Matrix representation of C62.29C23 in GL6(𝔽13)

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

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

C62.29C23 in GAP, Magma, Sage, TeX

`C_6^2._{29}C_2^3`
`% in TeX`

`G:=Group("C6^2.29C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,507);`
`// by ID`

`G=gap.SmallGroup(288,507);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,120,422,142,1356,9414]);`
`// Polycyclic`

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

׿
×
𝔽