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G = D8.10D6order 192 = 26·3

The non-split extension by D8 of D6 acting through Inn(D8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — D8.10D6
 Chief series C1 — C3 — C6 — C12 — C4×S3 — C4○D12 — Q8○D12 — D8.10D6
 Lower central C3 — C6 — C12 — D8.10D6
 Upper central C1 — C2 — C2×C4 — C4○D8

Generators and relations for D8.10D6
G = < a,b,c,d | a8=b2=c6=1, d2=a4, bab=a-1, ac=ca, ad=da, cbc-1=a4b, bd=db, dcd-1=a4c-1 >

Subgroups: 600 in 248 conjugacy classes, 99 normal (31 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), D8, SD16, SD16, Q16, Q16, C2×Q8, C4○D4, C4○D4, C3⋊C8, C24, Dic6, Dic6, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C8○D4, C2×Q16, C4○D8, C4○D8, C8.C22, 2- 1+4, S3×C8, C8⋊S3, Dic12, C4.Dic3, D4.S3, C3⋊Q16, C2×C24, C3×D8, C3×SD16, C3×Q16, C2×Dic6, C2×Dic6, C4○D12, C4○D12, D42S3, D42S3, S3×Q8, C3×C4○D4, Q8○D8, C8○D12, C2×Dic12, D83S3, D4.D6, S3×Q16, Q8.14D6, C3×C4○D8, Q8○D12, D8.10D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, S3×D4, S3×C23, Q8○D8, C2×S3×D4, D8.10D6

Smallest permutation representation of D8.10D6
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 63)(2 62)(3 61)(4 60)(5 59)(6 58)(7 57)(8 64)(9 80)(10 79)(11 78)(12 77)(13 76)(14 75)(15 74)(16 73)(17 34)(18 33)(19 40)(20 39)(21 38)(22 37)(23 36)(24 35)(25 46)(26 45)(27 44)(28 43)(29 42)(30 41)(31 48)(32 47)(49 90)(50 89)(51 96)(52 95)(53 94)(54 93)(55 92)(56 91)(65 85)(66 84)(67 83)(68 82)(69 81)(70 88)(71 87)(72 86)
(1 85 54 37 73 46)(2 86 55 38 74 47)(3 87 56 39 75 48)(4 88 49 40 76 41)(5 81 50 33 77 42)(6 82 51 34 78 43)(7 83 52 35 79 44)(8 84 53 36 80 45)(9 30 64 70 94 19)(10 31 57 71 95 20)(11 32 58 72 96 21)(12 25 59 65 89 22)(13 26 60 66 90 23)(14 27 61 67 91 24)(15 28 62 68 92 17)(16 29 63 69 93 18)
(1 46 5 42)(2 47 6 43)(3 48 7 44)(4 41 8 45)(9 66 13 70)(10 67 14 71)(11 68 15 72)(12 69 16 65)(17 92 21 96)(18 93 22 89)(19 94 23 90)(20 95 24 91)(25 59 29 63)(26 60 30 64)(27 61 31 57)(28 62 32 58)(33 54 37 50)(34 55 38 51)(35 56 39 52)(36 49 40 53)(73 85 77 81)(74 86 78 82)(75 87 79 83)(76 88 80 84)```

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 D6 D6 D6 S3×D4 S3×D4 Q8○D8 D8.10D6 kernel D8.10D6 C8○D12 C2×Dic12 D8⋊3S3 D4.D6 S3×Q16 Q8.14D6 C3×C4○D8 Q8○D12 C4○D8 Dic6 D12 C3⋊D4 C2×C8 D8 SD16 Q16 C4○D4 C4 C22 C3 C1 # reps 1 1 1 2 4 2 2 1 2 1 1 1 2 1 1 2 1 2 1 1 2 4

Matrix representation of D8.10D6 in GL4(𝔽73) generated by

 16 0 16 0 0 16 0 16 57 0 16 0 0 57 0 16
,
 42 11 31 62 62 31 11 42 31 62 31 62 11 42 11 42
,
 0 0 7 66 0 0 7 14 66 7 0 0 66 59 0 0
,
 0 0 66 7 0 0 14 7 7 66 0 0 59 66 0 0
`G:=sub<GL(4,GF(73))| [16,0,57,0,0,16,0,57,16,0,16,0,0,16,0,16],[42,62,31,11,11,31,62,42,31,11,31,11,62,42,62,42],[0,0,66,66,0,0,7,59,7,7,0,0,66,14,0,0],[0,0,7,59,0,0,66,66,66,14,0,0,7,7,0,0] >;`

D8.10D6 in GAP, Magma, Sage, TeX

`D_8._{10}D_6`
`% in TeX`

`G:=Group("D8.10D6");`
`// GroupNames label`

`G:=SmallGroup(192,1330);`
`// by ID`

`G=gap.SmallGroup(192,1330);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,387,184,570,185,136,438,235,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^6=1,d^2=a^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=a^4*c^-1>;`
`// generators/relations`

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