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## G = C4⋊C4.225D6order 192 = 26·3

### 3rd non-split extension by C4⋊C4 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C4⋊C4.225D6
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×C3⋊C8 — C2×C4.Dic3 — C4⋊C4.225D6
 Lower central C3 — C6 — C12 — C4⋊C4.225D6
 Upper central C1 — C22 — C22×C4 — C2×C4⋊C4

Generators and relations for C4⋊C4.225D6
G = < a,b,c,d | a4=b4=c6=1, d2=a-1b2, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a-1b-1, dcd-1=a2c-1 >

Subgroups: 248 in 118 conjugacy classes, 63 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C6 [×3], C6 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×6], C23, Dic3 [×2], C12 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4, C3⋊C8 [×4], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×4], C2×C12 [×4], C22×C6, C4.Q8 [×2], C2.D8 [×2], C2×C4⋊C4, C42⋊C2, C2×M4(2), C2×C3⋊C8 [×2], C4.Dic3 [×4], C4×Dic3, C4⋊Dic3 [×2], C6.D4, C3×C4⋊C4 [×2], C3×C4⋊C4, C22×C12, C22×C12, M4(2)⋊C4, C6.Q16 [×2], C12.Q8 [×2], C2×C4.Dic3, C23.26D6, C6×C4⋊C4, C4⋊C4.225D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, Dic6 [×2], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C2×C4⋊C4, C8⋊C22, C8.C22, Dic3⋊C4 [×4], C2×Dic6, S3×C2×C4, C2×C3⋊D4, M4(2)⋊C4, C2×Dic3⋊C4, D126C22, Q8.11D6, C4⋊C4.225D6

Smallest permutation representation of C4⋊C4.225D6
On 96 points
Generators in S96
```(1 71 5 67)(2 72 6 68)(3 65 7 69)(4 66 8 70)(9 88 13 84)(10 81 14 85)(11 82 15 86)(12 83 16 87)(17 45 21 41)(18 46 22 42)(19 47 23 43)(20 48 24 44)(25 60 29 64)(26 61 30 57)(27 62 31 58)(28 63 32 59)(33 93 37 89)(34 94 38 90)(35 95 39 91)(36 96 40 92)(49 79 53 75)(50 80 54 76)(51 73 55 77)(52 74 56 78)
(1 86 65 13)(2 85 66 12)(3 84 67 11)(4 83 68 10)(5 82 69 9)(6 81 70 16)(7 88 71 15)(8 87 72 14)(17 61 47 32)(18 60 48 31)(19 59 41 30)(20 58 42 29)(21 57 43 28)(22 64 44 27)(23 63 45 26)(24 62 46 25)(33 51 95 75)(34 50 96 74)(35 49 89 73)(36 56 90 80)(37 55 91 79)(38 54 92 78)(39 53 93 77)(40 52 94 76)
(1 37 22)(2 19 38 6 23 34)(3 39 24)(4 21 40 8 17 36)(5 33 18)(7 35 20)(9 75 31)(10 28 76 14 32 80)(11 77 25)(12 30 78 16 26 74)(13 79 27)(15 73 29)(41 92 70 45 96 66)(42 71 89)(43 94 72 47 90 68)(44 65 91)(46 67 93)(48 69 95)(49 58 88)(50 85 59 54 81 63)(51 60 82)(52 87 61 56 83 57)(53 62 84)(55 64 86)
(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)| (1,71,5,67)(2,72,6,68)(3,65,7,69)(4,66,8,70)(9,88,13,84)(10,81,14,85)(11,82,15,86)(12,83,16,87)(17,45,21,41)(18,46,22,42)(19,47,23,43)(20,48,24,44)(25,60,29,64)(26,61,30,57)(27,62,31,58)(28,63,32,59)(33,93,37,89)(34,94,38,90)(35,95,39,91)(36,96,40,92)(49,79,53,75)(50,80,54,76)(51,73,55,77)(52,74,56,78), (1,86,65,13)(2,85,66,12)(3,84,67,11)(4,83,68,10)(5,82,69,9)(6,81,70,16)(7,88,71,15)(8,87,72,14)(17,61,47,32)(18,60,48,31)(19,59,41,30)(20,58,42,29)(21,57,43,28)(22,64,44,27)(23,63,45,26)(24,62,46,25)(33,51,95,75)(34,50,96,74)(35,49,89,73)(36,56,90,80)(37,55,91,79)(38,54,92,78)(39,53,93,77)(40,52,94,76), (1,37,22)(2,19,38,6,23,34)(3,39,24)(4,21,40,8,17,36)(5,33,18)(7,35,20)(9,75,31)(10,28,76,14,32,80)(11,77,25)(12,30,78,16,26,74)(13,79,27)(15,73,29)(41,92,70,45,96,66)(42,71,89)(43,94,72,47,90,68)(44,65,91)(46,67,93)(48,69,95)(49,58,88)(50,85,59,54,81,63)(51,60,82)(52,87,61,56,83,57)(53,62,84)(55,64,86), (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( (1,71,5,67)(2,72,6,68)(3,65,7,69)(4,66,8,70)(9,88,13,84)(10,81,14,85)(11,82,15,86)(12,83,16,87)(17,45,21,41)(18,46,22,42)(19,47,23,43)(20,48,24,44)(25,60,29,64)(26,61,30,57)(27,62,31,58)(28,63,32,59)(33,93,37,89)(34,94,38,90)(35,95,39,91)(36,96,40,92)(49,79,53,75)(50,80,54,76)(51,73,55,77)(52,74,56,78), (1,86,65,13)(2,85,66,12)(3,84,67,11)(4,83,68,10)(5,82,69,9)(6,81,70,16)(7,88,71,15)(8,87,72,14)(17,61,47,32)(18,60,48,31)(19,59,41,30)(20,58,42,29)(21,57,43,28)(22,64,44,27)(23,63,45,26)(24,62,46,25)(33,51,95,75)(34,50,96,74)(35,49,89,73)(36,56,90,80)(37,55,91,79)(38,54,92,78)(39,53,93,77)(40,52,94,76), (1,37,22)(2,19,38,6,23,34)(3,39,24)(4,21,40,8,17,36)(5,33,18)(7,35,20)(9,75,31)(10,28,76,14,32,80)(11,77,25)(12,30,78,16,26,74)(13,79,27)(15,73,29)(41,92,70,45,96,66)(42,71,89)(43,94,72,47,90,68)(44,65,91)(46,67,93)(48,69,95)(49,58,88)(50,85,59,54,81,63)(51,60,82)(52,87,61,56,83,57)(53,62,84)(55,64,86), (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([(1,71,5,67),(2,72,6,68),(3,65,7,69),(4,66,8,70),(9,88,13,84),(10,81,14,85),(11,82,15,86),(12,83,16,87),(17,45,21,41),(18,46,22,42),(19,47,23,43),(20,48,24,44),(25,60,29,64),(26,61,30,57),(27,62,31,58),(28,63,32,59),(33,93,37,89),(34,94,38,90),(35,95,39,91),(36,96,40,92),(49,79,53,75),(50,80,54,76),(51,73,55,77),(52,74,56,78)], [(1,86,65,13),(2,85,66,12),(3,84,67,11),(4,83,68,10),(5,82,69,9),(6,81,70,16),(7,88,71,15),(8,87,72,14),(17,61,47,32),(18,60,48,31),(19,59,41,30),(20,58,42,29),(21,57,43,28),(22,64,44,27),(23,63,45,26),(24,62,46,25),(33,51,95,75),(34,50,96,74),(35,49,89,73),(36,56,90,80),(37,55,91,79),(38,54,92,78),(39,53,93,77),(40,52,94,76)], [(1,37,22),(2,19,38,6,23,34),(3,39,24),(4,21,40,8,17,36),(5,33,18),(7,35,20),(9,75,31),(10,28,76,14,32,80),(11,77,25),(12,30,78,16,26,74),(13,79,27),(15,73,29),(41,92,70,45,96,66),(42,71,89),(43,94,72,47,90,68),(44,65,91),(46,67,93),(48,69,95),(49,58,88),(50,85,59,54,81,63),(51,60,82),(52,87,61,56,83,57),(53,62,84),(55,64,86)], [(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)])`

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + - + + + - + - image C1 C2 C2 C2 C2 C2 C4 S3 D4 Q8 D4 D6 D6 Dic6 C4×S3 C3⋊D4 C3⋊D4 C8⋊C22 C8.C22 D12⋊6C22 Q8.11D6 kernel C4⋊C4.225D6 C6.Q16 C12.Q8 C2×C4.Dic3 C23.26D6 C6×C4⋊C4 C4.Dic3 C2×C4⋊C4 C2×C12 C2×C12 C22×C6 C4⋊C4 C22×C4 C2×C4 C2×C4 C2×C4 C23 C6 C6 C2 C2 # reps 1 2 2 1 1 1 8 1 1 2 1 2 1 4 4 2 2 1 1 2 2

Matrix representation of C4⋊C4.225D6 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 66 0 0 0 0 42 1 0 0 0 0 0 0 72 66 0 0 0 0 42 1
,
 52 8 0 0 0 0 36 21 0 0 0 0 0 0 51 15 0 0 0 0 65 22 0 0 0 0 0 0 5 50 0 0 0 0 17 68
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 64 0 0 0 0 0 0 64 0 0 0 0 0 0 65 0 0 0 0 0 0 65
,
 13 67 0 0 0 0 4 60 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 9 63 0 0 0 0 60 64 0 0

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,42,0,0,0,0,66,1,0,0,0,0,0,0,72,42,0,0,0,0,66,1],[52,36,0,0,0,0,8,21,0,0,0,0,0,0,51,65,0,0,0,0,15,22,0,0,0,0,0,0,5,17,0,0,0,0,50,68],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,65,0,0,0,0,0,0,65],[13,4,0,0,0,0,67,60,0,0,0,0,0,0,0,0,9,60,0,0,0,0,63,64,0,0,8,0,0,0,0,0,0,8,0,0] >;`

C4⋊C4.225D6 in GAP, Magma, Sage, TeX

`C_4\rtimes C_4._{225}D_6`
`% in TeX`

`G:=Group("C4:C4.225D6");`
`// GroupNames label`

`G:=SmallGroup(192,523);`
`// by ID`

`G=gap.SmallGroup(192,523);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,477,422,58,438,102,6278]);`
`// Polycyclic`

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

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