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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, (C2×C8)⋊17D6, (C2×Q8)⋊6D6, D6⋊C8⋊32C2, (C3×D4).9D4, C4.62(S3×D4), D6⋊3Q8⋊3C2, C12.47(C2×D4), (C6×Q8)⋊3C22, (C2×C24)⋊33C22, C6.57C22≀C2, (C2×SD16)⋊10S3, (C6×SD16)⋊20C2, (C2×D4).146D6, C2.D24⋊35C2, D4.8(C3⋊D4), C3⋊4(C22⋊SD16), C2.28(S3×SD16), C6.45(C2×SD16), D4⋊Dic3⋊33C2, C2.28(Q8⋊3D6), C6.78(C8⋊C22), C4⋊Dic3⋊20C22, (C2×Dic3).71D4, (C22×S3).91D4, (C6×D4).95C22, C22.266(S3×D4), C2.25(C23⋊2D6), (C2×C12).446C23, (C2×D12).120C22, (C2×S3×D4).6C2, (C2×C3⋊C8)⋊8C22, C4.42(C2×C3⋊D4), (C2×C6).358(C2×D4), (S3×C2×C4).47C22, (C2×Q8⋊2S3)⋊17C2, (C2×C4).535(C22×S3), SmallGroup(192,728)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C12 — D6⋊6SD16
Chief series
C1 — C3 — C6 — C2×C6 — C2×C12 — S3×C2×C4 — C2×S3×D4 — D6⋊6SD16
Lower central
C3 — C6 — C2×C12 — D6⋊6SD16
Upper central
C1 — C22 — C2×C4 — C2×SD16

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, C2×C4, C2×C4, D4, D4, Q8, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C3⋊C8, C24, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×S3, C22×C6, C22⋊C8, D4⋊C4, C22⋊Q8, C2×SD16, C2×SD16, C22×D4, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, D6⋊C4, Q8⋊2S3, C2×C24, C3×SD16, S3×C2×C4, C2×D12, S3×D4, C2×C3⋊D4, C6×D4, C6×Q8, S3×C23, C22⋊SD16, D6⋊C8, C2.D24, D4⋊Dic3, C2×Q8⋊2S3, D6⋊3Q8, C6×SD16, C2×S3×D4, D6⋊6SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C3⋊D4, C22×S3, C22≀C2, C2×SD16, C8⋊C22, S3×D4, C2×C3⋊D4, C22⋊SD16, S3×SD16, 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⋊C22S3×D4S3×D4S3×SD16Q8⋊3D6
kernelD6⋊6SD16D6⋊C8C2.D24D4⋊Dic3C2×Q8⋊2S3D6⋊3Q8C6×SD16C2×S3×D4C2×SD16D12C2×Dic3C3×D4C22×S3C2×C8C2×D4C2×Q8D6D4C6C4C22C2C2
# 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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