Copied to
clipboard

## G = D12⋊16D4order 192 = 26·3

### 4th semidirect product of D12 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D12⋊16D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C22×D12 — D12⋊16D4
 Lower central C3 — C6 — C2×C12 — D12⋊16D4
 Upper central C1 — C22 — C22×C4 — C4⋊D4

Generators and relations for D1216D4
G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, cac-1=a7, ad=da, cbc-1=a3b, bd=db, dcd=c-1 >

Subgroups: 752 in 198 conjugacy classes, 47 normal (27 characteristic)
C1, C2 [×3], C2 [×7], C3, C4 [×2], C4 [×2], C22, C22 [×2], C22 [×21], S3 [×4], C6 [×3], C6 [×3], C8 [×2], C2×C4 [×2], C2×C4 [×3], D4 [×14], C23, C23 [×11], C12 [×2], C12 [×2], D6 [×16], C2×C6, C2×C6 [×2], C2×C6 [×5], C22⋊C4, C4⋊C4, C2×C8 [×2], D8 [×4], C22×C4, C2×D4, C2×D4 [×8], C24, C3⋊C8 [×2], D12 [×4], D12 [×6], C2×C12 [×2], C2×C12 [×3], C3×D4 [×4], C22×S3 [×10], C22×C6, C22×C6, C22⋊C8, D4⋊C4 [×2], C4⋊D4, C2×D8 [×2], C22×D4, C2×C3⋊C8 [×2], D4⋊S3 [×4], C3×C22⋊C4, C3×C4⋊C4, C2×D12 [×2], C2×D12 [×5], C22×C12, C6×D4, C6×D4, S3×C23, C22⋊D8, C6.D8 [×2], C12.55D4, C2×D4⋊S3 [×2], C3×C4⋊D4, C22×D12, D1216D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], D8 [×2], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C22≀C2, C2×D8, C8⋊C22, D4⋊S3 [×2], S3×D4 [×2], C2×C3⋊D4, C22⋊D8, C2×D4⋊S3, C232D6, D4⋊D6, D1216D4

Smallest permutation representation of D1216D4
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 43)(2 42)(3 41)(4 40)(5 39)(6 38)(7 37)(8 48)(9 47)(10 46)(11 45)(12 44)(13 35)(14 34)(15 33)(16 32)(17 31)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(24 36)
(1 16 44 36)(2 23 45 31)(3 18 46 26)(4 13 47 33)(5 20 48 28)(6 15 37 35)(7 22 38 30)(8 17 39 25)(9 24 40 32)(10 19 41 27)(11 14 42 34)(12 21 43 29)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 27)(14 28)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 25)(24 26)(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), (1,43)(2,42)(3,41)(4,40)(5,39)(6,38)(7,37)(8,48)(9,47)(10,46)(11,45)(12,44)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(24,36), (1,16,44,36)(2,23,45,31)(3,18,46,26)(4,13,47,33)(5,20,48,28)(6,15,37,35)(7,22,38,30)(8,17,39,25)(9,24,40,32)(10,19,41,27)(11,14,42,34)(12,21,43,29), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,25)(24,26)(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), (1,43)(2,42)(3,41)(4,40)(5,39)(6,38)(7,37)(8,48)(9,47)(10,46)(11,45)(12,44)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(24,36), (1,16,44,36)(2,23,45,31)(3,18,46,26)(4,13,47,33)(5,20,48,28)(6,15,37,35)(7,22,38,30)(8,17,39,25)(9,24,40,32)(10,19,41,27)(11,14,42,34)(12,21,43,29), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,25)(24,26)(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)], [(1,43),(2,42),(3,41),(4,40),(5,39),(6,38),(7,37),(8,48),(9,47),(10,46),(11,45),(12,44),(13,35),(14,34),(15,33),(16,32),(17,31),(18,30),(19,29),(20,28),(21,27),(22,26),(23,25),(24,36)], [(1,16,44,36),(2,23,45,31),(3,18,46,26),(4,13,47,33),(5,20,48,28),(6,15,37,35),(7,22,38,30),(8,17,39,25),(9,24,40,32),(10,19,41,27),(11,14,42,34),(12,21,43,29)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,27),(14,28),(15,29),(16,30),(17,31),(18,32),(19,33),(20,34),(21,35),(22,36),(23,25),(24,26),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)])

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 D6 D8 C3⋊D4 C3⋊D4 C8⋊C22 S3×D4 D4⋊S3 D4⋊D6 kernel D12⋊16D4 C6.D8 C12.55D4 C2×D4⋊S3 C3×C4⋊D4 C22×D12 C4⋊D4 D12 C2×C12 C22×C6 C4⋊C4 C22×C4 C2×D4 C2×C6 C2×C4 C23 C6 C4 C22 C2 # reps 1 2 1 2 1 1 1 4 1 1 1 1 1 4 2 2 1 2 2 2

Matrix representation of D1216D4 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 0 0 1 0 0 0 0 72 0
,
 1 0 0 0 0 0 0 72 0 0 0 0 0 0 1 72 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 71 0 0 0 0 37 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 57 57 0 0 0 0 57 16
,
 1 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,37,0,0,0,0,71,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,57,57,0,0,0,0,57,16],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;

D1216D4 in GAP, Magma, Sage, TeX

D_{12}\rtimes_{16}D_4
% in TeX

G:=Group("D12:16D4");
// GroupNames label

G:=SmallGroup(192,595);
// by ID

G=gap.SmallGroup(192,595);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,1123,297,136,6278]);
// 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=a^7,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽