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## G = C4○D4⋊C12order 192 = 26·3

### The semidirect product of C4○D4 and C12 acting via C12/C2=C6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C4○D4⋊C12
 Chief series C1 — C2 — Q8 — C2×Q8 — C2×SL2(𝔽3) — C2×C4.A4 — C4○D4⋊C12
 Lower central Q8 — C4○D4⋊C12
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C4○D4⋊C12
G = < a,b,c,d | a4=c2=d12=1, b2=a2, dcd-1=ab=ba, ac=ca, dad-1=a-1, cbc=a2b, dbd-1=a-1bc >

Subgroups: 253 in 84 conjugacy classes, 27 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C12, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, SL2(𝔽3), C2×C12, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C2×C4○D4, C3×C4⋊C4, C2×SL2(𝔽3), C4.A4, C23.33C23, C4×SL2(𝔽3), C2×C4.A4, C4○D4⋊C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C12, A4, C2×C6, C2×C12, C2×A4, C4×A4, C22×A4, C2×C4×A4, Q8.A4, D4.A4, C4○D4⋊C12

Smallest permutation representation of C4○D4⋊C12
On 64 points
Generators in S64
```(1 15 10 6)(2 7 11 16)(3 13 12 8)(4 5 9 14)(17 62 34 42)(18 43 35 63)(19 64 36 44)(20 45 37 53)(21 54 38 46)(22 47 39 55)(23 56 40 48)(24 49 29 57)(25 58 30 50)(26 51 31 59)(27 60 32 52)(28 41 33 61)
(1 39 10 22)(2 36 11 19)(3 33 12 28)(4 30 9 25)(5 50 14 58)(6 47 15 55)(7 44 16 64)(8 41 13 61)(17 21 34 38)(18 31 35 26)(20 24 37 29)(23 27 40 32)(42 46 62 54)(43 59 63 51)(45 49 53 57)(48 52 56 60)
(1 51)(2 56)(3 45)(4 62)(5 34)(6 26)(7 40)(8 20)(9 42)(10 59)(11 48)(12 53)(13 37)(14 17)(15 31)(16 23)(18 55)(19 60)(21 50)(22 43)(24 61)(25 54)(27 44)(28 49)(29 41)(30 46)(32 64)(33 57)(35 47)(36 52)(38 58)(39 63)
(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)```

`G:=sub<Sym(64)| (1,15,10,6)(2,7,11,16)(3,13,12,8)(4,5,9,14)(17,62,34,42)(18,43,35,63)(19,64,36,44)(20,45,37,53)(21,54,38,46)(22,47,39,55)(23,56,40,48)(24,49,29,57)(25,58,30,50)(26,51,31,59)(27,60,32,52)(28,41,33,61), (1,39,10,22)(2,36,11,19)(3,33,12,28)(4,30,9,25)(5,50,14,58)(6,47,15,55)(7,44,16,64)(8,41,13,61)(17,21,34,38)(18,31,35,26)(20,24,37,29)(23,27,40,32)(42,46,62,54)(43,59,63,51)(45,49,53,57)(48,52,56,60), (1,51)(2,56)(3,45)(4,62)(5,34)(6,26)(7,40)(8,20)(9,42)(10,59)(11,48)(12,53)(13,37)(14,17)(15,31)(16,23)(18,55)(19,60)(21,50)(22,43)(24,61)(25,54)(27,44)(28,49)(29,41)(30,46)(32,64)(33,57)(35,47)(36,52)(38,58)(39,63), (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)>;`

`G:=Group( (1,15,10,6)(2,7,11,16)(3,13,12,8)(4,5,9,14)(17,62,34,42)(18,43,35,63)(19,64,36,44)(20,45,37,53)(21,54,38,46)(22,47,39,55)(23,56,40,48)(24,49,29,57)(25,58,30,50)(26,51,31,59)(27,60,32,52)(28,41,33,61), (1,39,10,22)(2,36,11,19)(3,33,12,28)(4,30,9,25)(5,50,14,58)(6,47,15,55)(7,44,16,64)(8,41,13,61)(17,21,34,38)(18,31,35,26)(20,24,37,29)(23,27,40,32)(42,46,62,54)(43,59,63,51)(45,49,53,57)(48,52,56,60), (1,51)(2,56)(3,45)(4,62)(5,34)(6,26)(7,40)(8,20)(9,42)(10,59)(11,48)(12,53)(13,37)(14,17)(15,31)(16,23)(18,55)(19,60)(21,50)(22,43)(24,61)(25,54)(27,44)(28,49)(29,41)(30,46)(32,64)(33,57)(35,47)(36,52)(38,58)(39,63), (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) );`

`G=PermutationGroup([[(1,15,10,6),(2,7,11,16),(3,13,12,8),(4,5,9,14),(17,62,34,42),(18,43,35,63),(19,64,36,44),(20,45,37,53),(21,54,38,46),(22,47,39,55),(23,56,40,48),(24,49,29,57),(25,58,30,50),(26,51,31,59),(27,60,32,52),(28,41,33,61)], [(1,39,10,22),(2,36,11,19),(3,33,12,28),(4,30,9,25),(5,50,14,58),(6,47,15,55),(7,44,16,64),(8,41,13,61),(17,21,34,38),(18,31,35,26),(20,24,37,29),(23,27,40,32),(42,46,62,54),(43,59,63,51),(45,49,53,57),(48,52,56,60)], [(1,51),(2,56),(3,45),(4,62),(5,34),(6,26),(7,40),(8,20),(9,42),(10,59),(11,48),(12,53),(13,37),(14,17),(15,31),(16,23),(18,55),(19,60),(21,50),(22,43),(24,61),(25,54),(27,44),(28,49),(29,41),(30,46),(32,64),(33,57),(35,47),(36,52),(38,58),(39,63)], [(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)]])`

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 4 4 4 4 type + + + + + + - image C1 C2 C2 C3 C4 C6 C6 C12 A4 C2×A4 C4×A4 Q8.A4 Q8.A4 D4.A4 D4.A4 kernel C4○D4⋊C12 C4×SL2(𝔽3) C2×C4.A4 C23.33C23 C4.A4 C4×Q8 C2×C4○D4 C4○D4 C4⋊C4 C2×C4 C4 C2 C2 C2 C2 # reps 1 2 1 2 4 4 2 8 1 3 4 1 2 1 2

Matrix representation of C4○D4⋊C12 in GL7(𝔽13)

 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 8
,
 0 1 0 0 0 0 0 1 0 0 0 0 0 0 12 12 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 1 0
,
 12 12 12 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 6 2 0 0 0 0 0 2 7 0 0 0 0 0 0 0 7 11 0 0 0 0 0 11 6
,
 5 0 0 0 0 0 0 0 0 5 0 0 0 0 8 8 8 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 10 9 0 0 0 1 0 0 0 0 0 0 10 9 0 0

`G:=sub<GL(7,GF(13))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,8],[0,1,12,0,0,0,0,1,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,0],[12,0,0,0,0,0,0,12,0,1,0,0,0,0,12,1,0,0,0,0,0,0,0,0,6,2,0,0,0,0,0,2,7,0,0,0,0,0,0,0,7,11,0,0,0,0,0,11,6],[5,0,8,0,0,0,0,0,0,8,0,0,0,0,0,5,8,0,0,0,0,0,0,0,0,0,1,10,0,0,0,0,0,0,9,0,0,0,1,10,0,0,0,0,0,0,9,0,0] >;`

C4○D4⋊C12 in GAP, Magma, Sage, TeX

`C_4\circ D_4\rtimes C_{12}`
`% in TeX`

`G:=Group("C4oD4:C12");`
`// GroupNames label`

`G:=SmallGroup(192,999);`
`// by ID`

`G=gap.SmallGroup(192,999);`
`# by ID`

`G:=PCGroup([7,-2,-2,-3,-2,-2,2,-2,672,1373,92,438,172,775,285,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=c^2=d^12=1,b^2=a^2,d*c*d^-1=a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c=a^2*b,d*b*d^-1=a^-1*b*c>;`
`// generators/relations`

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