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## G = C4⋊C4.150D6order 192 = 26·3

### 23rd non-split extension by C4⋊C4 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C4⋊C4.150D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — S3×C2×C4 — C2×Q8⋊3S3 — C4⋊C4.150D6
 Lower central C3 — C6 — C12 — C4⋊C4.150D6
 Upper central C1 — C22 — C2×C4 — Q8⋊C4

Generators and relations for C4⋊C4.150D6
G = < a,b,c,d | a4=b4=d2=1, c6=a2, bab-1=cac-1=dad=a-1, cbc-1=a-1b-1, dbd=ab-1, dcd=a2c5 >

Subgroups: 456 in 158 conjugacy classes, 55 normal (37 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×6], C22, C22 [×8], S3 [×4], C6 [×3], C8 [×2], C2×C4, C2×C4 [×12], D4 [×7], Q8 [×2], Q8, C23 [×2], Dic3 [×2], Dic3, C12 [×2], C12 [×3], D6 [×2], D6 [×6], C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8 [×3], C22×C4 [×2], C2×D4 [×2], C2×Q8, C4○D4 [×6], C3⋊C8, C24, C4×S3 [×4], C4×S3 [×4], D12 [×2], D12 [×5], C2×Dic3, C2×Dic3, C2×C12, C2×C12 [×2], C3×Q8 [×2], C3×Q8, C22×S3, C22×S3, D4⋊C4 [×2], Q8⋊C4, Q8⋊C4, C42⋊C2, C22×C8, C2×C4○D4, S3×C8 [×2], C2×C3⋊C8, C4×Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, Q83S3 [×4], Q83S3 [×2], C6×Q8, C23.24D4, C6.D8, C2.D24, Q82Dic3, C3×Q8⋊C4, C4⋊C47S3, S3×C2×C8, C2×Q83S3, C4⋊C4.150D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], C22×S3, C2×C22⋊C4, C4○D8 [×2], S3×C2×C4, S3×D4 [×2], C23.24D4, S3×C22⋊C4, Q8.7D6, D24⋊C2, C4⋊C4.150D6

Smallest permutation representation of C4⋊C4.150D6
On 96 points
Generators in S96
```(1 52 7 58)(2 59 8 53)(3 54 9 60)(4 49 10 55)(5 56 11 50)(6 51 12 57)(13 74 19 80)(14 81 20 75)(15 76 21 82)(16 83 22 77)(17 78 23 84)(18 73 24 79)(25 66 31 72)(26 61 32 67)(27 68 33 62)(28 63 34 69)(29 70 35 64)(30 65 36 71)(37 90 43 96)(38 85 44 91)(39 92 45 86)(40 87 46 93)(41 94 47 88)(42 89 48 95)
(1 43 80 65)(2 25 81 85)(3 45 82 67)(4 27 83 87)(5 47 84 69)(6 29 73 89)(7 37 74 71)(8 31 75 91)(9 39 76 61)(10 33 77 93)(11 41 78 63)(12 35 79 95)(13 30 52 90)(14 44 53 66)(15 32 54 92)(16 46 55 68)(17 34 56 94)(18 48 57 70)(19 36 58 96)(20 38 59 72)(21 26 60 86)(22 40 49 62)(23 28 50 88)(24 42 51 64)
(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 9)(2 8)(3 7)(4 6)(10 12)(13 15)(16 24)(17 23)(18 22)(19 21)(25 44)(26 43)(27 42)(28 41)(29 40)(30 39)(31 38)(32 37)(33 48)(34 47)(35 46)(36 45)(49 57)(50 56)(51 55)(52 54)(58 60)(61 90)(62 89)(63 88)(64 87)(65 86)(66 85)(67 96)(68 95)(69 94)(70 93)(71 92)(72 91)(73 83)(74 82)(75 81)(76 80)(77 79)```

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

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

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

42 conjugacy classes

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

42 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 C4 S3 D4 D4 D4 D6 D6 D6 C4×S3 C4○D8 S3×D4 S3×D4 Q8.7D6 D24⋊C2 kernel C4⋊C4.150D6 C6.D8 C2.D24 Q8⋊2Dic3 C3×Q8⋊C4 C4⋊C4⋊7S3 S3×C2×C8 C2×Q8⋊3S3 Q8⋊3S3 Q8⋊C4 C4×S3 C2×Dic3 C22×S3 C4⋊C4 C2×C8 C2×Q8 Q8 C6 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 1 1 1 1 1 4 8 1 1 2 2

Matrix representation of C4⋊C4.150D6 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 0 72 0 0 1 0
,
 46 0 0 0 0 46 0 0 0 0 16 16 0 0 16 57
,
 1 1 0 0 72 0 0 0 0 0 0 46 0 0 46 0
,
 72 72 0 0 0 1 0 0 0 0 72 0 0 0 0 1
`G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,0,1,0,0,72,0],[46,0,0,0,0,46,0,0,0,0,16,16,0,0,16,57],[1,72,0,0,1,0,0,0,0,0,0,46,0,0,46,0],[72,0,0,0,72,1,0,0,0,0,72,0,0,0,0,1] >;`

C4⋊C4.150D6 in GAP, Magma, Sage, TeX

`C_4\rtimes C_4._{150}D_6`
`% in TeX`

`G:=Group("C4:C4.150D6");`
`// GroupNames label`

`G:=SmallGroup(192,363);`
`// by ID`

`G=gap.SmallGroup(192,363);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,232,219,58,570,136,851,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^6=a^2,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^-1*b^-1,d*b*d=a*b^-1,d*c*d=a^2*c^5>;`
`// generators/relations`

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