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G = D12.31D4order 192 = 26·3

1st non-split extension by D12 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D12.31D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C22×D12 — D12.31D4
 Lower central C3 — C6 — C2×C12 — D12.31D4
 Upper central C1 — C22 — C22×C4 — C22⋊C8

Generators and relations for D12.31D4
G = < a,b,c,d | a12=b2=d2=1, c4=a6, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd=a3c3 >

Subgroups: 736 in 188 conjugacy classes, 47 normal (25 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×2], C4 [×3], C22, C22 [×2], C22 [×18], S3 [×4], C6 [×3], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], D4 [×10], Q8 [×2], C23, C23 [×10], Dic3 [×2], C12 [×2], C12, D6 [×16], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], C22×C4, C2×D4 [×7], C2×Q8, C24, C24 [×2], Dic6 [×2], D12 [×4], D12 [×6], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×2], C22×S3 [×10], C22×C6, C22⋊C8, D4⋊C4 [×2], C22⋊Q8, C2×SD16 [×2], C22×D4, C24⋊C2 [×4], Dic3⋊C4, C4⋊Dic3, C6.D4, C2×C24 [×2], C2×Dic6, C2×D12 [×2], C2×D12 [×5], C22×C12, S3×C23, C22⋊SD16, C2.D24 [×2], C3×C22⋊C8, C2×C24⋊C2 [×2], C12.48D4, C22×D12, D12.31D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], SD16 [×2], C2×D4 [×3], D12 [×2], C22×S3, C22≀C2, C2×SD16, C8⋊C22, C24⋊C2 [×2], C2×D12, S3×D4 [×2], C22⋊SD16, D6⋊D4, C2×C24⋊C2, C8⋊D6, D12.31D4

Smallest permutation representation of D12.31D4
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)(14 24)(15 23)(16 22)(17 21)(18 20)(25 31)(26 30)(27 29)(32 36)(33 35)(37 38)(39 48)(40 47)(41 46)(42 45)(43 44)
(1 30 47 18 7 36 41 24)(2 31 48 19 8 25 42 13)(3 32 37 20 9 26 43 14)(4 33 38 21 10 27 44 15)(5 34 39 22 11 28 45 16)(6 35 40 23 12 29 46 17)
(13 34)(14 35)(15 36)(16 25)(17 26)(18 27)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)```

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 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 3 4 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 2 2 2 4 24 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 SD16 D12 D12 C24⋊C2 C8⋊C22 S3×D4 C8⋊D6 kernel D12.31D4 C2.D24 C3×C22⋊C8 C2×C24⋊C2 C12.48D4 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 D12.31D4 in GL4(𝔽73) generated by

 66 7 0 0 66 59 0 0 0 0 72 0 0 0 0 72
,
 66 7 0 0 14 7 0 0 0 0 72 0 0 0 12 1
,
 48 11 0 0 62 37 0 0 0 0 61 71 0 0 35 12
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 61 72
`G:=sub<GL(4,GF(73))| [66,66,0,0,7,59,0,0,0,0,72,0,0,0,0,72],[66,14,0,0,7,7,0,0,0,0,72,12,0,0,0,1],[48,62,0,0,11,37,0,0,0,0,61,35,0,0,71,12],[1,0,0,0,0,1,0,0,0,0,1,61,0,0,0,72] >;`

D12.31D4 in GAP, Magma, Sage, TeX

`D_{12}._{31}D_4`
`% in TeX`

`G:=Group("D12.31D4");`
`// GroupNames label`

`G:=SmallGroup(192,290);`
`// by ID`

`G=gap.SmallGroup(192,290);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,58,1123,136,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=d^2=1,c^4=a^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d=a^3*c^3>;`
`// generators/relations`

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