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G = C4○D12⋊C4order 192 = 26·3

3rd semidirect product of C4○D12 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C4○D12⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×D12 — C2×C4○D12 — C4○D12⋊C4
 Lower central C3 — C6 — C12 — C4○D12⋊C4
 Upper central C1 — C22 — C22×C4 — C2×C4⋊C4

Generators and relations for C4○D12⋊C4
G = < a,b,c,d | a4=c2=d4=1, b6=a2, ab=ba, ac=ca, dad-1=a-1, cbc=a2b5, dbd-1=a2b, dcd-1=b3c >

Subgroups: 440 in 162 conjugacy classes, 63 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×6], S3 [×2], C6 [×3], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×9], D4 [×7], Q8 [×3], C23, C23, Dic3 [×2], C12 [×2], C12 [×2], C12 [×2], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×Q8, C4○D4 [×6], C3⋊C8 [×2], Dic6 [×2], Dic6, C4×S3 [×4], D12 [×2], D12, C2×Dic3, C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×4], C2×C12 [×4], C22×S3, C22×C6, D4⋊C4 [×2], Q8⋊C4 [×2], C2×C4⋊C4, C2×M4(2), C2×C4○D4, C2×C3⋊C8 [×2], C4.Dic3 [×2], C3×C4⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12 [×4], C4○D12 [×2], C2×C3⋊D4, C22×C12, C22×C12, C23.36D4, C6.D8 [×2], C6.SD16 [×2], C2×C4.Dic3, C6×C4⋊C4, C2×C4○D12, C4○D12⋊C4
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], D12 [×2], C3⋊D4 [×2], C22×S3, C2×C22⋊C4, C8⋊C22, C8.C22, D6⋊C4 [×4], S3×C2×C4, C2×D12, C2×C3⋊D4, C23.36D4, C2×D6⋊C4, D126C22, Q8.11D6, C4○D12⋊C4

Smallest permutation representation of C4○D12⋊C4
On 96 points
Generators in S96
```(1 60 7 54)(2 49 8 55)(3 50 9 56)(4 51 10 57)(5 52 11 58)(6 53 12 59)(13 62 19 68)(14 63 20 69)(15 64 21 70)(16 65 22 71)(17 66 23 72)(18 67 24 61)(25 79 31 73)(26 80 32 74)(27 81 33 75)(28 82 34 76)(29 83 35 77)(30 84 36 78)(37 93 43 87)(38 94 44 88)(39 95 45 89)(40 96 46 90)(41 85 47 91)(42 86 48 92)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 18)(14 17)(15 16)(19 24)(20 23)(21 22)(25 33)(26 32)(27 31)(28 30)(34 36)(37 39)(40 48)(41 47)(42 46)(43 45)(49 58)(50 57)(51 56)(52 55)(53 54)(59 60)(61 68)(62 67)(63 66)(64 65)(69 72)(70 71)(73 81)(74 80)(75 79)(76 78)(82 84)(85 91)(86 90)(87 89)(92 96)(93 95)
(1 46 16 25)(2 41 17 32)(3 48 18 27)(4 43 19 34)(5 38 20 29)(6 45 21 36)(7 40 22 31)(8 47 23 26)(9 42 24 33)(10 37 13 28)(11 44 14 35)(12 39 15 30)(49 91 66 80)(50 86 67 75)(51 93 68 82)(52 88 69 77)(53 95 70 84)(54 90 71 79)(55 85 72 74)(56 92 61 81)(57 87 62 76)(58 94 63 83)(59 89 64 78)(60 96 65 73)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A ··· 6G 8A 8B 8C 8D 12A ··· 12L order 1 2 2 2 2 2 2 2 3 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 12 12 2 2 2 2 2 4 4 4 4 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 4 4 4 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 S3 D4 D4 D6 D6 C4×S3 D12 C3⋊D4 C3⋊D4 C8⋊C22 C8.C22 D12⋊6C22 Q8.11D6 kernel C4○D12⋊C4 C6.D8 C6.SD16 C2×C4.Dic3 C6×C4⋊C4 C2×C4○D12 C4○D12 C2×C4⋊C4 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 3 1 2 1 4 4 2 2 1 1 2 2

Matrix representation of C4○D12⋊C4 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 30 13 0 0 0 0 60 43 0 0 43 60 0 0 0 0 13 30 0 0
,
 72 1 0 0 0 0 72 0 0 0 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 1 72 0 0 0 0 1 0 0 0
,
 72 0 0 0 0 0 72 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0
,
 46 0 0 0 0 0 0 46 0 0 0 0 0 0 56 32 15 41 0 0 41 15 32 56 0 0 15 41 17 41 0 0 32 56 32 58

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,43,13,0,0,0,0,60,30,0,0,30,60,0,0,0,0,13,43,0,0],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,72,72,0,0,0,0,1,0,0,0],[72,72,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,56,41,15,32,0,0,32,15,41,56,0,0,15,32,17,32,0,0,41,56,41,58] >;`

C4○D12⋊C4 in GAP, Magma, Sage, TeX

`C_4\circ D_{12}\rtimes C_4`
`% in TeX`

`G:=Group("C4oD12:C4");`
`// GroupNames label`

`G:=SmallGroup(192,525);`
`// by ID`

`G=gap.SmallGroup(192,525);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,422,387,58,1684,438,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=c^2=d^4=1,b^6=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c=a^2*b^5,d*b*d^-1=a^2*b,d*c*d^-1=b^3*c>;`
`// generators/relations`

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