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G = D12:16D4  order 192 = 26·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D12:16D4, C4:C4:3D6, (C2xC6):2D8, (C2xD4):1D6, C4:D4:1S3, C4.98(S3xD4), C6.54(C2xD8), C3:3(C22:D8), (C2xC12).71D4, (C6xD4):1C22, C6.44C22wrC2, C22:3(D4:S3), C6.D8:33C2, C12.145(C2xD4), (C22xC6).82D4, (C22xD12):13C2, (C22xC4).135D6, C2.12(C23:2D6), C2.12(D4:D6), C12.55D4:10C2, C6.114(C8:C22), (C2xC12).355C23, C23.65(C3:D4), (C2xD12).240C22, (C22xC12).159C22, (C2xD4:S3):8C2, (C2xC3:C8):5C22, C2.9(C2xD4:S3), (C3xC4:D4):1C2, (C3xC4:C4):5C22, (C2xC6).486(C2xD4), (C2xC4).49(C3:D4), (C2xC4).455(C22xS3), C22.161(C2xC3:D4), SmallGroup(192,595)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2xC12 — D12:16D4
Chief series
C1 — C3 — C6 — C12 — C2xC12 — C2xD12 — C22xD12 — D12:16D4
Lower central
C3 — C6 — C2xC12 — D12:16D4
Upper central
C1 — C22 — C22xC4 — C4:D4

Generators and relations for D12:16D4
 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, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, C23, C23, C12, C12, D6, C2xC6, C2xC6, C2xC6, C22:C4, C4:C4, C2xC8, D8, C22xC4, C2xD4, C2xD4, C24, C3:C8, D12, D12, C2xC12, C2xC12, C3xD4, C22xS3, C22xC6, C22xC6, C22:C8, D4:C4, C4:D4, C2xD8, C22xD4, C2xC3:C8, D4:S3, C3xC22:C4, C3xC4:C4, C2xD12, C2xD12, C22xC12, C6xD4, C6xD4, S3xC23, C22:D8, C6.D8, C12.55D4, C2xD4:S3, C3xC4:D4, C22xD12, D12:16D4
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2xD4, C3:D4, C22xS3, C22wrC2, C2xD8, C8:C22, D4:S3, S3xD4, C2xC3:D4, C22:D8, C2xD4:S3, C23:2D6, D4:D6, D12:16D4

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

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

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

33 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J 3 4A4B4C4D6A6B6C6D6E6F6G8A8B8C8D12A12B12C12D12E12F
order122222222223444466666668888121212121212
size11112281212121222248222448812121212444488

33 irreducible representations

dim11111122222222224444
type++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D4D6D6D6D8C3:D4C3:D4C8:C22S3xD4D4:S3D4:D6
kernelD12:16D4C6.D8C12.55D4C2xD4:S3C3xC4:D4C22xD12C4:D4D12C2xC12C22xC6C4:C4C22xC4C2xD4C2xC6C2xC4C23C6C4C22C2
# reps12121114111114221222

Matrix representation of D12:16D4 ►in GL6(F73)

7200000
0720000
0007200
0017200
000001
0000720
,
100000
0720000
0017200
0007200
000001
000010
,
0710000
3700000
001000
000100
00005757
00005716
,
100000
0720000
001000
000100
0000720
0000072

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] >;
 

D12:16D4 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
 

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