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## G = D12⋊13D4order 192 = 26·3

### 1st semidirect product of D12 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D12⋊13D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C22×D12 — D12⋊13D4
 Lower central C3 — C6 — C2×C12 — D12⋊13D4
 Upper central C1 — C22 — C22×C4 — C22⋊C8

Generators and relations for D1213D4
G = < a,b,c,d | a12=b2=c4=d2=1, bab=cac-1=a-1, ad=da, cbc-1=a7b, bd=db, dcd=c-1 >

Subgroups: 832 in 198 conjugacy classes, 47 normal (25 characteristic)
C1, C2 [×3], C2 [×7], C3, C4 [×2], C4 [×2], C22, C22 [×2], C22 [×21], S3 [×5], C6 [×3], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×3], D4 [×14], C23, C23 [×11], Dic3, C12 [×2], C12, D6 [×19], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4, C4⋊C4, C2×C8 [×2], D8 [×4], C22×C4, C2×D4 [×9], C24, C24 [×2], D12 [×4], D12 [×8], C2×Dic3, C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×2], C22×S3 [×11], C22×C6, C22⋊C8, D4⋊C4 [×2], C4⋊D4, C2×D8 [×2], C22×D4, D24 [×4], C4⋊Dic3, D6⋊C4, C2×C24 [×2], C2×D12, C2×D12 [×2], C2×D12 [×5], C2×C3⋊D4, C22×C12, S3×C23, C22⋊D8, C2.D24 [×2], C3×C22⋊C8, C2×D24 [×2], C127D4, C22×D12, D1213D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], D8 [×2], C2×D4 [×3], D12 [×2], C22×S3, C22≀C2, C2×D8, C8⋊C22, D24 [×2], C2×D12, S3×D4 [×2], C22⋊D8, D6⋊D4, C2×D24, C8⋊D6, D1213D4

Smallest permutation representation of D1213D4
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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 23)(14 22)(15 21)(16 20)(17 19)(25 30)(26 29)(27 28)(31 36)(32 35)(33 34)(37 43)(38 42)(39 41)(44 48)(45 47)
(1 20 34 48)(2 19 35 47)(3 18 36 46)(4 17 25 45)(5 16 26 44)(6 15 27 43)(7 14 28 42)(8 13 29 41)(9 24 30 40)(10 23 31 39)(11 22 32 38)(12 21 33 37)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 25)(11 26)(12 27)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)```

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 D8 D12 D12 D24 C8⋊C22 S3×D4 C8⋊D6 kernel D12⋊13D4 C2.D24 C3×C22⋊C8 C2×D24 C12⋊7D4 C22×D12 C22⋊C8 D12 C2×C12 C22×C6 C2×C8 C22×C4 C2×C6 C2×C4 C23 C22 C6 C4 C2 # reps 1 2 1 2 1 1 1 4 1 1 2 1 4 2 2 8 1 2 2

Matrix representation of D1213D4 in GL4(𝔽73) generated by

 66 7 0 0 66 59 0 0 0 0 1 0 0 0 0 1
,
 66 7 0 0 14 7 0 0 0 0 1 0 0 0 0 1
,
 50 68 0 0 18 23 0 0 0 0 0 1 0 0 72 0
,
 72 0 0 0 0 72 0 0 0 0 72 0 0 0 0 1
`G:=sub<GL(4,GF(73))| [66,66,0,0,7,59,0,0,0,0,1,0,0,0,0,1],[66,14,0,0,7,7,0,0,0,0,1,0,0,0,0,1],[50,18,0,0,68,23,0,0,0,0,0,72,0,0,1,0],[72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,1] >;`

D1213D4 in GAP, Magma, Sage, TeX

`D_{12}\rtimes_{13}D_4`
`% in TeX`

`G:=Group("D12:13D4");`
`// GroupNames label`

`G:=SmallGroup(192,291);`
`// by ID`

`G=gap.SmallGroup(192,291);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,226,1123,136,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^7*b,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

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