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G = D66SD16order 192 = 26·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D66SD16, D12.16D4, (C2×C8)⋊17D6, (C2×Q8)⋊6D6, D6⋊C832C2, (C3×D4).9D4, C4.62(S3×D4), D63Q83C2, C12.47(C2×D4), (C6×Q8)⋊3C22, (C2×C24)⋊33C22, C6.57C22≀C2, (C2×SD16)⋊10S3, (C6×SD16)⋊20C2, (C2×D4).146D6, C2.D2435C2, D4.8(C3⋊D4), C34(C22⋊SD16), C2.28(S3×SD16), C6.45(C2×SD16), D4⋊Dic333C2, C2.28(Q83D6), C6.78(C8⋊C22), C4⋊Dic320C22, (C2×Dic3).71D4, (C22×S3).91D4, (C6×D4).95C22, C22.266(S3×D4), C2.25(C232D6), (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×Q82S3)⋊17C2, (C2×C4).535(C22×S3), SmallGroup(192,728)

Series: Derived Chief Lower central Upper central

C1C2×C12 — D66SD16
C1C3C6C2×C6C2×C12S3×C2×C4C2×S3×D4 — D66SD16
C3C6C2×C12 — D66SD16
C1C22C2×C4C2×SD16

Generators and relations for D66SD16
 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, Q82S3, 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×Q82S3, D63Q8, C6×SD16, C2×S3×D4, D66SD16
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, Q83D6, C232D6, D66SD16

Smallest permutation representation of D66SD16
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×SD16Q83D6
kernelD66SD16D6⋊C8C2.D24D4⋊Dic3C2×Q82S3D63Q8C6×SD16C2×S3×D4C2×SD16D12C2×Dic3C3×D4C22×S3C2×C8C2×D4C2×Q8D6D4C6C4C22C2C2
# reps11111111121211114411122

Matrix representation of D66SD16 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] >;

D66SD16 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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