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## G = D12⋊21D4order 192 = 26·3

### 9th semidirect product of D12 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — D12⋊21D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22⋊C4 — D12⋊21D4
 Lower central C3 — C2×C6 — D12⋊21D4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for D1221D4
G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, cac-1=dad=a5, cbc-1=dbd=a10b, dcd=c-1 >

Subgroups: 1040 in 334 conjugacy classes, 107 normal (43 characteristic)
C1, C2 [×3], C2 [×9], C3, C4 [×2], C4 [×8], C22, C22 [×2], C22 [×27], S3 [×7], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×13], D4 [×18], Q8 [×2], C23, C23 [×15], Dic3 [×3], C12 [×2], C12 [×5], D6 [×4], D6 [×21], C2×C6, C2×C6 [×2], C2×C6 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×2], C4⋊C4, C22×C4, C22×C4 [×5], C2×D4 [×13], C2×Q8, C4○D4 [×4], C24 [×2], C4×S3 [×8], D12 [×4], D12 [×12], C2×Dic3 [×3], C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×4], C2×C12 [×2], C3×Q8 [×2], C22×S3, C22×S3 [×4], C22×S3 [×10], C22×C6, C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, C4×Dic3, Dic3⋊C4, D6⋊C4, D6⋊C4 [×8], C6.D4, C3×C22⋊C4 [×2], C3×C4⋊C4, C3×C4⋊C4 [×2], S3×C2×C4, S3×C2×C4 [×4], C2×D12 [×2], C2×D12 [×6], C2×D12 [×4], Q83S3 [×4], C2×C3⋊D4, C22×C12, C6×Q8, S3×C23 [×2], D45D4, S3×C22⋊C4 [×2], D6⋊D4 [×2], Dic35D4, D6.D4 [×2], C12⋊D4, C12⋊D4 [×2], C4×C3⋊D4, C12.23D4, C3×C22⋊Q8, C22×D12, C2×Q83S3, D1221D4
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×2], C24, C22×S3 [×7], C22×D4, C2×C4○D4, 2+ 1+4, S3×D4 [×2], Q83S3 [×2], S3×C23, D45D4, C2×S3×D4, C2×Q83S3, D4○D12, D1221D4

Smallest permutation representation of D1221D4
On 48 points
Generators in S48
```(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)
(1 3)(4 12)(5 11)(6 10)(7 9)(13 21)(14 20)(15 19)(16 18)(22 24)(25 33)(26 32)(27 31)(28 30)(34 36)(37 39)(40 48)(41 47)(42 46)(43 45)
(1 40 22 25)(2 45 23 30)(3 38 24 35)(4 43 13 28)(5 48 14 33)(6 41 15 26)(7 46 16 31)(8 39 17 36)(9 44 18 29)(10 37 19 34)(11 42 20 27)(12 47 21 32)
(1 31)(2 36)(3 29)(4 34)(5 27)(6 32)(7 25)(8 30)(9 35)(10 28)(11 33)(12 26)(13 37)(14 42)(15 47)(16 40)(17 45)(18 38)(19 43)(20 48)(21 41)(22 46)(23 39)(24 44)```

`G:=sub<Sym(48)| (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,21)(14,20)(15,19)(16,18)(22,24)(25,33)(26,32)(27,31)(28,30)(34,36)(37,39)(40,48)(41,47)(42,46)(43,45), (1,40,22,25)(2,45,23,30)(3,38,24,35)(4,43,13,28)(5,48,14,33)(6,41,15,26)(7,46,16,31)(8,39,17,36)(9,44,18,29)(10,37,19,34)(11,42,20,27)(12,47,21,32), (1,31)(2,36)(3,29)(4,34)(5,27)(6,32)(7,25)(8,30)(9,35)(10,28)(11,33)(12,26)(13,37)(14,42)(15,47)(16,40)(17,45)(18,38)(19,43)(20,48)(21,41)(22,46)(23,39)(24,44)>;`

`G:=Group( (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,21)(14,20)(15,19)(16,18)(22,24)(25,33)(26,32)(27,31)(28,30)(34,36)(37,39)(40,48)(41,47)(42,46)(43,45), (1,40,22,25)(2,45,23,30)(3,38,24,35)(4,43,13,28)(5,48,14,33)(6,41,15,26)(7,46,16,31)(8,39,17,36)(9,44,18,29)(10,37,19,34)(11,42,20,27)(12,47,21,32), (1,31)(2,36)(3,29)(4,34)(5,27)(6,32)(7,25)(8,30)(9,35)(10,28)(11,33)(12,26)(13,37)(14,42)(15,47)(16,40)(17,45)(18,38)(19,43)(20,48)(21,41)(22,46)(23,39)(24,44) );`

`G=PermutationGroup([(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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,21),(14,20),(15,19),(16,18),(22,24),(25,33),(26,32),(27,31),(28,30),(34,36),(37,39),(40,48),(41,47),(42,46),(43,45)], [(1,40,22,25),(2,45,23,30),(3,38,24,35),(4,43,13,28),(5,48,14,33),(6,41,15,26),(7,46,16,31),(8,39,17,36),(9,44,18,29),(10,37,19,34),(11,42,20,27),(12,47,21,32)], [(1,31),(2,36),(3,29),(4,34),(5,27),(6,32),(7,25),(8,30),(9,35),(10,28),(11,33),(12,26),(13,37),(14,42),(15,47),(16,40),(17,45),(18,38),(19,43),(20,48),(21,41),(22,46),(23,39),(24,44)])`

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 D6 C4○D4 2+ 1+4 S3×D4 Q8⋊3S3 D4○D12 kernel D12⋊21D4 S3×C22⋊C4 D6⋊D4 Dic3⋊5D4 D6.D4 C12⋊D4 C4×C3⋊D4 C12.23D4 C3×C22⋊Q8 C22×D12 C2×Q8⋊3S3 C22⋊Q8 D12 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C2×C6 C6 C4 C22 C2 # reps 1 2 2 1 2 3 1 1 1 1 1 1 4 2 3 1 1 4 1 2 2 2

Matrix representation of D1221D4 in GL6(𝔽13)

 0 1 0 0 0 0 12 0 0 0 0 0 0 0 1 1 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 8 0 0 0 0 5 0 0 0 0 0 0 0 12 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 12 0
,
 0 5 0 0 0 0 8 0 0 0 0 0 0 0 12 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0

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

D1221D4 in GAP, Magma, Sage, TeX

`D_{12}\rtimes_{21}D_4`
`% in TeX`

`G:=Group("D12:21D4");`
`// GroupNames label`

`G:=SmallGroup(192,1189);`
`// by ID`

`G=gap.SmallGroup(192,1189);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,184,1571,297,192,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^5,c*b*c^-1=d*b*d=a^10*b,d*c*d=c^-1>;`
`// generators/relations`

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