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## G = D6.4C42order 192 = 26·3

### 4th non-split extension by D6 of C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — D6.4C42
 Chief series C1 — C3 — C6 — C12 — C2×C12 — S3×C2×C4 — C42⋊2S3 — D6.4C42
 Lower central C3 — C6 — D6.4C42
 Upper central C1 — C2×C4 — C8⋊C4

Generators and relations for D6.4C42
G = < a,b,c,d | a6=b2=c4=1, d4=a3, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=a3c >

Subgroups: 248 in 130 conjugacy classes, 75 normal (33 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, C2×C4, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C3⋊C8, C24, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C4×C8, C8⋊C4, C8⋊C4, C42⋊C2, C22×C8, C2×M4(2), S3×C8, C8⋊S3, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, D6⋊C4, C4×C12, C2×C24, S3×C2×C4, C82M4(2), C4×C3⋊C8, C8×Dic3, C24⋊C4, C3×C8⋊C4, C422S3, S3×C2×C8, C2×C8⋊S3, D6.4C42
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C42, C22×C4, C4×S3, C22×S3, C2×C42, C8○D4, S3×C2×C4, C82M4(2), S3×C42, D12.C4, D6.4C42

Smallest permutation representation of D6.4C42
On 96 points
Generators in S96
```(1 19 54 5 23 50)(2 20 55 6 24 51)(3 21 56 7 17 52)(4 22 49 8 18 53)(9 65 93 13 69 89)(10 66 94 14 70 90)(11 67 95 15 71 91)(12 68 96 16 72 92)(25 39 86 29 35 82)(26 40 87 30 36 83)(27 33 88 31 37 84)(28 34 81 32 38 85)(41 78 59 45 74 63)(42 79 60 46 75 64)(43 80 61 47 76 57)(44 73 62 48 77 58)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 49)(9 93)(10 94)(11 95)(12 96)(13 89)(14 90)(15 91)(16 92)(17 21)(18 22)(19 23)(20 24)(33 84)(34 85)(35 86)(36 87)(37 88)(38 81)(39 82)(40 83)(41 63)(42 64)(43 57)(44 58)(45 59)(46 60)(47 61)(48 62)(73 77)(74 78)(75 79)(76 80)
(1 86 46 11)(2 83 47 16)(3 88 48 13)(4 85 41 10)(5 82 42 15)(6 87 43 12)(7 84 44 9)(8 81 45 14)(17 27 73 65)(18 32 74 70)(19 29 75 67)(20 26 76 72)(21 31 77 69)(22 28 78 66)(23 25 79 71)(24 30 80 68)(33 62 93 52)(34 59 94 49)(35 64 95 54)(36 61 96 51)(37 58 89 56)(38 63 90 53)(39 60 91 50)(40 57 92 55)
(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)```

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4N 6A 6B 6C 8A ··· 8H 8I ··· 8P 8Q 8R 8S 8T 12A 12B 12C 12D 12E 12F 12G 12H 24A ··· 24H order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 ··· 4 6 6 6 8 ··· 8 8 ··· 8 8 8 8 8 12 12 12 12 12 12 12 12 24 ··· 24 size 1 1 1 1 6 6 2 1 1 1 1 2 2 2 2 6 ··· 6 2 2 2 2 ··· 2 3 ··· 3 6 6 6 6 2 2 2 2 4 4 4 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 S3 D6 D6 C4×S3 C4×S3 C8○D4 D12.C4 kernel D6.4C42 C4×C3⋊C8 C8×Dic3 C24⋊C4 C3×C8⋊C4 C42⋊2S3 S3×C2×C8 C2×C8⋊S3 S3×C8 C8⋊S3 Dic3⋊C4 D6⋊C4 C8⋊C4 C42 C2×C8 C8 C2×C4 C6 C2 # reps 1 1 1 1 1 1 1 1 8 8 4 4 1 1 2 8 4 8 4

Matrix representation of D6.4C42 in GL5(𝔽73)

 1 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 72 72 0 0 0 1 0
,
 72 0 0 0 0 0 72 0 0 0 0 0 1 0 0 0 0 0 72 72 0 0 0 0 1
,
 46 0 0 0 0 0 0 1 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 1
,
 46 0 0 0 0 0 63 0 0 0 0 0 10 0 0 0 0 0 27 0 0 0 0 0 27

`G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72,1,0,0,0,72,0],[72,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,72,1],[46,0,0,0,0,0,0,72,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[46,0,0,0,0,0,63,0,0,0,0,0,10,0,0,0,0,0,27,0,0,0,0,0,27] >;`

D6.4C42 in GAP, Magma, Sage, TeX

`D_6._4C_4^2`
`% in TeX`

`G:=Group("D6.4C4^2");`
`// GroupNames label`

`G:=SmallGroup(192,267);`
`// by ID`

`G=gap.SmallGroup(192,267);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,120,387,58,136,6278]);`
`// Polycyclic`

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

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