Copied to
clipboard

G = D12⋊D4order 192 = 26·3

6th semidirect product of D12 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D65D8, D126D4, (C2×C8)⋊3D6, (C2×D8)⋊4S3, (C2×D4)⋊3D6, (C3×D4)⋊5D4, D6⋊C817C2, (C6×D8)⋊13C2, C6.45(C2×D8), C4.59(S3×D4), C2.28(S3×D8), D63D43C2, C34(C22⋊D8), D42(C3⋊D4), C12.46(C2×D4), (C6×D4)⋊3C22, (C2×C24)⋊19C22, C6.55C22≀C2, C2.D2417C2, D4⋊Dic328C2, C2.29(D8⋊S3), C6.50(C8⋊C22), C4⋊Dic319C22, (C2×Dic3).65D4, (C22×S3).90D4, C22.256(S3×D4), C2.23(C232D6), (C2×C12).433C23, (C2×D12).116C22, (C2×S3×D4)⋊2C2, (C2×C3⋊C8)⋊7C22, (C2×D4⋊S3)⋊19C2, C4.36(C2×C3⋊D4), (C2×C6).346(C2×D4), (S3×C2×C4).44C22, (C2×C4).523(C22×S3), SmallGroup(192,715)

Series: Derived Chief Lower central Upper central

C1C2×C12 — D12⋊D4
C1C3C6C2×C6C2×C12S3×C2×C4C2×S3×D4 — D12⋊D4
C3C6C2×C12 — D12⋊D4
C1C22C2×C4C2×D8

Generators and relations for D12⋊D4
 G = < a,b,c,d | a12=b2=c4=d2=1, bab=cac-1=a-1, dad=a5, cbc-1=a7b, dbd=a4b, dcd=c-1 >

Subgroups: 760 in 198 conjugacy classes, 45 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C2×D4, C24, C3⋊C8, C24, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×D4, C22×S3, C22×S3, C22×C6, C22⋊C8, D4⋊C4, C4⋊D4, C2×D8, C2×D8, C22×D4, C2×C3⋊C8, C4⋊Dic3, D4⋊S3, C6.D4, C2×C24, C3×D8, S3×C2×C4, C2×D12, S3×D4, C2×C3⋊D4, C6×D4, S3×C23, C22⋊D8, D6⋊C8, C2.D24, D4⋊Dic3, C2×D4⋊S3, D63D4, C6×D8, C2×S3×D4, D12⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C3⋊D4, C22×S3, C22≀C2, C2×D8, C8⋊C22, S3×D4, C2×C3⋊D4, C22⋊D8, S3×D8, D8⋊S3, C232D6, D12⋊D4

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

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

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

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

33 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J 3 4A4B4C4D6A6B6C6D6E6F6G8A8B8C8D12A12B24A24B24C24D
order122222222223444466666668888121224242424
size111144668121222212242228888441212444444

33 irreducible representations

dim1111111122222222244444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D4D4D6D6D8C3⋊D4C8⋊C22S3×D4S3×D4S3×D8D8⋊S3
kernelD12⋊D4D6⋊C8C2.D24D4⋊Dic3C2×D4⋊S3D63D4C6×D8C2×S3×D4C2×D8D12C2×Dic3C3×D4C22×S3C2×C8C2×D4D6D4C6C4C22C2C2
# reps1111111112121124411122

Matrix representation of D12⋊D4 in GL6(𝔽73)

100000
010000
00727200
001000
000013
00004872
,
7200000
0720000
00727200
000100
0000720
0000251
,
41710000
38320000
0072000
001100
0000025
0000380
,
100000
41720000
001000
00727200
000010
000001

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,48,0,0,0,0,3,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,72,1,0,0,0,0,0,0,72,25,0,0,0,0,0,1],[41,38,0,0,0,0,71,32,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,0,38,0,0,0,0,25,0],[1,41,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

D12⋊D4 in GAP, Magma, Sage, TeX

D_{12}\rtimes D_4
% in TeX

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

G:=SmallGroup(192,715);
// by ID

G=gap.SmallGroup(192,715);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,851,438,102,6278]);
// 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,d*a*d=a^5,c*b*c^-1=a^7*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽