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G = D65SD16order 192 = 26·3

1st semidirect product of D6 and SD16 acting via SD16/D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D65SD16, D12.8D4, D4.6D12, C4⋊C42D6, D6⋊C89C2, (C2×C8)⋊16D6, (C3×D4).1D4, C12.1(C2×D4), C4.85(S3×D4), C4.2(C2×D12), C4.D121C2, C6.D87C2, D4⋊C410S3, (C2×C24)⋊15C22, C6.20C22≀C2, (C2×D4).135D6, C32(C22⋊SD16), C6.23(C2×SD16), C2.11(S3×SD16), C2.13(D8⋊S3), C6.31(C8⋊C22), (C2×Dic3).20D4, (C22×S3).72D4, (C6×D4).37C22, C22.174(S3×D4), C2.23(D6⋊D4), (C2×C12).216C23, (C2×Dic6)⋊13C22, (C2×D12).50C22, (C2×S3×D4).5C2, (C2×C3⋊C8)⋊3C22, (C3×C4⋊C4)⋊4C22, (C2×D4.S3)⋊3C2, (C2×C24⋊C2)⋊14C2, (S3×C2×C4).9C22, (C2×C6).229(C2×D4), (C3×D4⋊C4)⋊10C2, (C2×C4).323(C22×S3), SmallGroup(192,335)

Series: Derived Chief Lower central Upper central

C1C2×C12 — D65SD16
C1C3C6C12C2×C12S3×C2×C4C2×S3×D4 — D65SD16
C3C6C2×C12 — D65SD16
C1C22C2×C4D4⋊C4

Generators and relations for D65SD16
 G = < a,b,c,d | a6=b2=c8=d2=1, bab=cac-1=a-1, ad=da, cbc-1=ab, bd=db, dcd=c3 >

Subgroups: 712 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, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C3⋊C8, C24, Dic6, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C22×S3, C22×S3, C22×C6, C22⋊C8, D4⋊C4, D4⋊C4, C22⋊Q8, C2×SD16, C22×D4, C24⋊C2, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, D4.S3, C3×C4⋊C4, C2×C24, C2×Dic6, S3×C2×C4, C2×D12, S3×D4, C2×C3⋊D4, C6×D4, S3×C23, C22⋊SD16, C6.D8, D6⋊C8, C3×D4⋊C4, C4.D12, C2×C24⋊C2, C2×D4.S3, C2×S3×D4, D65SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, D12, C22×S3, C22≀C2, C2×SD16, C8⋊C22, C2×D12, S3×D4, C22⋊SD16, D6⋊D4, D8⋊S3, S3×SD16, D65SD16

Smallest permutation representation of D65SD16
On 48 points
Generators in S48
(1 40 26 22 42 12)(2 13 43 23 27 33)(3 34 28 24 44 14)(4 15 45 17 29 35)(5 36 30 18 46 16)(6 9 47 19 31 37)(7 38 32 20 48 10)(8 11 41 21 25 39)
(1 16)(2 47)(3 10)(4 41)(5 12)(6 43)(7 14)(8 45)(9 13)(11 15)(17 39)(18 26)(19 33)(20 28)(21 35)(22 30)(23 37)(24 32)(25 29)(27 31)(34 48)(36 42)(38 44)(40 46)
(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 22)(2 17)(3 20)(4 23)(5 18)(6 21)(7 24)(8 19)(9 25)(10 28)(11 31)(12 26)(13 29)(14 32)(15 27)(16 30)(33 45)(34 48)(35 43)(36 46)(37 41)(38 44)(39 47)(40 42)

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

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

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

33 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E6A6B6C6D6E8A8B8C8D12A12B12C12D24A24B24C24D
order12222222223444446666688881212121224242424
size111144661212222812242228844121244884444

33 irreducible representations

dim11111111222222222244444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D4D4D6D6D6SD16D12C8⋊C22S3×D4S3×D4D8⋊S3S3×SD16
kernelD65SD16C6.D8D6⋊C8C3×D4⋊C4C4.D12C2×C24⋊C2C2×D4.S3C2×S3×D4D4⋊C4D12C2×Dic3C3×D4C22×S3C4⋊C4C2×C8C2×D4D6D4C6C4C22C2C2
# reps11111111121211114411122

Matrix representation of D65SD16 in GL4(𝔽73) generated by

0100
72100
0010
0001
,
72100
0100
00720
00072
,
59700
661400
006767
00667
,
72000
07200
0010
00072
G:=sub<GL(4,GF(73))| [0,72,0,0,1,1,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,1,1,0,0,0,0,72,0,0,0,0,72],[59,66,0,0,7,14,0,0,0,0,67,6,0,0,67,67],[72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,72] >;

D65SD16 in GAP, Magma, Sage, TeX

D_6\rtimes_5{\rm SD}_{16}
% in TeX

G:=Group("D6:5SD16");
// GroupNames label

G:=SmallGroup(192,335);
// by ID

G=gap.SmallGroup(192,335);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,422,135,268,570,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=c^8=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a*b,b*d=d*b,d*c*d=c^3>;
// generators/relations

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