Copied to
clipboard

## G = C3⋊C8.29D4order 192 = 26·3

### 6th non-split extension by C3⋊C8 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C3⋊C8.29D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×Dic6 — C12.48D4 — C3⋊C8.29D4
 Lower central C3 — C6 — C2×C12 — C3⋊C8.29D4
 Upper central C1 — C22 — C22×C4 — C22⋊Q8

Generators and relations for C3⋊C8.29D4
G = < a,b,c,d | a3=b8=c4=d2=1, bab-1=cac-1=a-1, ad=da, cbc-1=b-1, bd=db, dcd=b4c-1 >

Subgroups: 272 in 114 conjugacy classes, 45 normal (39 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C6 [×3], C6 [×2], C8 [×3], C2×C4 [×2], C2×C4 [×6], Q8 [×4], C23, Dic3 [×2], C12 [×2], C12 [×3], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×3], C2×C8 [×4], Q16 [×2], C22×C4, C2×Q8, C2×Q8, C3⋊C8 [×2], C3⋊C8, Dic6 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×4], C3×Q8 [×2], C22×C6, Q8⋊C4 [×2], C2.D8, C22⋊Q8, C22⋊Q8, C22×C8, C2×Q16, C2×C3⋊C8 [×2], C2×C3⋊C8 [×2], Dic3⋊C4, C4⋊Dic3, C3⋊Q16 [×2], C6.D4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C22×C12, C6×Q8, C8.18D4, C6.Q16, C6.SD16, Q82Dic3, C22×C3⋊C8, C12.48D4, C2×C3⋊Q16, C3×C22⋊Q8, C3⋊C8.29D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], Q16 [×2], C2×D4 [×2], C4○D4, C3⋊D4 [×2], C22×S3, C4⋊D4, C2×Q16, C4○D8, C3⋊Q16 [×2], S3×D4, D42S3, C2×C3⋊D4, C8.18D4, C23.14D6, C2×C3⋊Q16, Q8.13D6, C3⋊C8.29D4

Smallest permutation representation of C3⋊C8.29D4
On 96 points
Generators in S96
```(1 83 94)(2 95 84)(3 85 96)(4 89 86)(5 87 90)(6 91 88)(7 81 92)(8 93 82)(9 30 49)(10 50 31)(11 32 51)(12 52 25)(13 26 53)(14 54 27)(15 28 55)(16 56 29)(17 64 33)(18 34 57)(19 58 35)(20 36 59)(21 60 37)(22 38 61)(23 62 39)(24 40 63)(41 75 70)(42 71 76)(43 77 72)(44 65 78)(45 79 66)(46 67 80)(47 73 68)(48 69 74)
(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 47 12 24)(2 46 13 23)(3 45 14 22)(4 44 15 21)(5 43 16 20)(6 42 9 19)(7 41 10 18)(8 48 11 17)(25 40 94 73)(26 39 95 80)(27 38 96 79)(28 37 89 78)(29 36 90 77)(30 35 91 76)(31 34 92 75)(32 33 93 74)(49 58 88 71)(50 57 81 70)(51 64 82 69)(52 63 83 68)(53 62 84 67)(54 61 85 66)(55 60 86 65)(56 59 87 72)
(17 44)(18 45)(19 46)(20 47)(21 48)(22 41)(23 42)(24 43)(33 78)(34 79)(35 80)(36 73)(37 74)(38 75)(39 76)(40 77)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 72)(64 65)```

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 8A ··· 8H 12A 12B 12C 12D 12E 12F 12G 12H order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 6 6 6 6 6 8 ··· 8 12 12 12 12 12 12 12 12 size 1 1 1 1 2 2 2 2 2 2 2 8 8 24 24 2 2 2 4 4 6 ··· 6 4 4 4 4 8 8 8 8

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + - + - - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 D6 C4○D4 Q16 C3⋊D4 C3⋊D4 C4○D8 S3×D4 D4⋊2S3 C3⋊Q16 Q8.13D6 kernel C3⋊C8.29D4 C6.Q16 C6.SD16 Q8⋊2Dic3 C22×C3⋊C8 C12.48D4 C2×C3⋊Q16 C3×C22⋊Q8 C22⋊Q8 C3⋊C8 C2×C12 C22×C6 C4⋊C4 C22×C4 C2×Q8 C12 C2×C6 C2×C4 C23 C6 C4 C4 C22 C2 # reps 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 2 4 2 2 4 1 1 2 2

Matrix representation of C3⋊C8.29D4 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 72 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 63 0 0 0 0 0 5 51 0 0 0 0 0 0 66 7 0 0 0 0 14 7 0 0 0 0 0 0 27 0 0 0 0 0 0 46
,
 64 7 0 0 0 0 72 9 0 0 0 0 0 0 7 66 0 0 0 0 59 66 0 0 0 0 0 0 0 27 0 0 0 0 27 0
,
 1 0 0 0 0 0 13 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[63,5,0,0,0,0,0,51,0,0,0,0,0,0,66,14,0,0,0,0,7,7,0,0,0,0,0,0,27,0,0,0,0,0,0,46],[64,72,0,0,0,0,7,9,0,0,0,0,0,0,7,59,0,0,0,0,66,66,0,0,0,0,0,0,0,27,0,0,0,0,27,0],[1,13,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;`

C3⋊C8.29D4 in GAP, Magma, Sage, TeX

`C_3\rtimes C_8._{29}D_4`
`% in TeX`

`G:=Group("C3:C8.29D4");`
`// GroupNames label`

`G:=SmallGroup(192,610);`
`// by ID`

`G=gap.SmallGroup(192,610);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,219,184,1123,297,136,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^3=b^8=c^4=d^2=1,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d=b^4*c^-1>;`
`// generators/relations`

׿
×
𝔽