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G = D6:6SD16  order 192 = 26·3

2nd semidirect product of D6 and SD16 acting via SD16/D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6:6SD16, D12.16D4, (C2xC8):17D6, (C2xQ8):6D6, D6:C8:32C2, (C3xD4).9D4, C4.62(S3xD4), D6:3Q8:3C2, C12.47(C2xD4), (C6xQ8):3C22, (C2xC24):33C22, C6.57C22wrC2, (C2xSD16):10S3, (C6xSD16):20C2, (C2xD4).146D6, C2.D24:35C2, D4.8(C3:D4), C3:4(C22:SD16), C2.28(S3xSD16), C6.45(C2xSD16), D4:Dic3:33C2, C2.28(Q8:3D6), C6.78(C8:C22), C4:Dic3:20C22, (C2xDic3).71D4, (C22xS3).91D4, (C6xD4).95C22, C22.266(S3xD4), C2.25(C23:2D6), (C2xC12).446C23, (C2xD12).120C22, (C2xS3xD4).6C2, (C2xC3:C8):8C22, C4.42(C2xC3:D4), (C2xC6).358(C2xD4), (S3xC2xC4).47C22, (C2xQ8:2S3):17C2, (C2xC4).535(C22xS3), SmallGroup(192,728)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2xC12 — D6:6SD16
Chief series
C1 — C3 — C6 — C2xC6 — C2xC12 — S3xC2xC4 — C2xS3xD4 — D6:6SD16
Lower central
C3 — C6 — C2xC12 — D6:6SD16
Upper central
C1 — C22 — C2xC4 — C2xSD16

Generators and relations for D6:6SD16
 G = < a,b,c,d | a6=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd=c3 >

Subgroups: 696 in 188 conjugacy classes, 45 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, D4, Q8, C23, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C22:C4, C4:C4, C2xC8, C2xC8, SD16, C22xC4, C2xD4, C2xD4, C2xQ8, C24, C3:C8, C24, C4xS3, D12, D12, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C3xD4, C3xQ8, C22xS3, C22xS3, C22xC6, C22:C8, D4:C4, C22:Q8, C2xSD16, C2xSD16, C22xD4, C2xC3:C8, Dic3:C4, C4:Dic3, D6:C4, Q8:2S3, C2xC24, C3xSD16, S3xC2xC4, C2xD12, S3xD4, C2xC3:D4, C6xD4, C6xQ8, S3xC23, C22:SD16, D6:C8, C2.D24, D4:Dic3, C2xQ8:2S3, D6:3Q8, C6xSD16, C2xS3xD4, D6:6SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2xD4, C3:D4, C22xS3, C22wrC2, C2xSD16, C8:C22, S3xD4, C2xC3:D4, C22:SD16, S3xSD16, Q8:3D6, C23:2D6, D6:6SD16

Smallest permutation representation of D6:6SD16
►On 48 points
Generators in S48
(1 24 47 28 13 33)(2 17 48 29 14 34)(3 18 41 30 15 35)(4 19 42 31 16 36)(5 20 43 32 9 37)(6 21 44 25 10 38)(7 22 45 26 11 39)(8 23 46 27 12 40)

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

(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)

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

33 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E6A6B6C6D6E8A8B8C8D12A12B12C12D24A24B24C24D
order12222222223444446666688881212121224242424
size111144661212222812242228844121244884444

33 irreducible representations

dim11111111222222222244444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D4D4D6D6D6SD16C3:D4C8:C22S3xD4S3xD4S3xSD16Q8:3D6
kernelD6:6SD16D6:C8C2.D24D4:Dic3C2xQ8:2S3D6:3Q8C6xSD16C2xS3xD4C2xSD16D12C2xDic3C3xD4C22xS3C2xC8C2xD4C2xQ8D6D4C6C4C22C2C2
# reps11111111121211114411122

Matrix representation of D6:6SD16 ►in GL6(F73)

7200000
0720000
001000
000100
0000072
0000172
,
7200000
010000
001000
000100
0000172
0000072
,
010000
100000
00676700
0066700
0000720
0000072
,
100000
010000
001000
0007200
000010
000001

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,72,72],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,67,6,0,0,0,0,67,67,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

D6:6SD16 in GAP, Magma, Sage, TeX 

D_6\rtimes_6{\rm SD}_{16}
 
% in TeX
 
G:=Group("D6:6SD16");
 
// GroupNames label
 
G:=SmallGroup(192,728);
 
// by ID
 
G=gap.SmallGroup(192,728);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,184,851,438,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^2=c^8=d^2=1,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=c^3>;
 
// generators/relations
 

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