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G = SD16⋊13D6  order 192 = 26·3

2nd semidirect product of SD16 and D6 acting through Inn(SD16)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: SD16⋊13D6, D12.29D4, C12.9C24, D24⋊21C22, C24.36C23, Dic6.29D4, D12.5C23, Dic6.5C23, Dic12⋊18C22, (C2×C8)⋊11D6, C8○D12⋊5C2, C4○D24⋊8C2, Q8⋊3D6⋊5C2, (C2×Q8)⋊14D6, C4.76(S3×D4), C3⋊D4.9D4, C3⋊C8.3C23, D4⋊6D6⋊6C2, (S3×C8)⋊9C22, (C2×C24)⋊6C22, D4⋊S3⋊2C22, (C2×SD16)⋊6S3, (S3×SD16)⋊5C2, (C6×SD16)⋊2C2, D6.27(C2×D4), D4.D6⋊5C2, C3⋊2(D4○SD16), C12.84(C2×D4), C4.9(S3×C23), (S3×Q8)⋊1C22, Q8.7D6⋊5C2, D12⋊6C22⋊8C2, (C2×D4).116D6, C4○D12⋊4C22, (C4×S3).5C23, C8.12(C22×S3), D4.S3⋊2C22, (C6×Q8)⋊19C22, C3⋊Q16⋊1C22, D4.7(C22×S3), (C3×D4).7C23, (S3×D4).1C22, C22.21(S3×D4), C8⋊S3⋊10C22, C24⋊C2⋊19C22, (C3×Q8).3C23, Q8.11D6⋊7C2, Q8.15D6⋊3C2, Dic3.32(C2×D4), Q8⋊3S3⋊1C22, Q8⋊2S3⋊1C22, C6.110(C22×D4), Q8.13(C22×S3), (C2×C12).526C23, (C3×SD16)⋊14C22, D4⋊2S3.1C22, (C6×D4).167C22, C4.Dic3⋊29C22, C2.83(C2×S3×D4), (C2×C6).399(C2×D4), (C2×C4).230(C22×S3), SmallGroup(192,1321)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — SD16⋊13D6
Chief series
C1 — C3 — C6 — C12 — C4×S3 — C4○D12 — D4⋊6D6 — SD16⋊13D6
Lower central
C3 — C6 — C12 — SD16⋊13D6
Upper central
C1 — C2 — C2×C4 — C2×SD16

Generators and relations for SD16⋊13D6
 G = < a,b,c,d | a8=b2=c6=d2=1, bab=cac-1=a3, ad=da, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 712 in 258 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), D8, SD16, SD16, Q16, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×S3, C22×C6, C8○D4, C2×SD16, C2×SD16, C4○D8, C8⋊C22, C8.C22, 2+ 1+4, 2- 1+4, S3×C8, C8⋊S3, C24⋊C2, D24, Dic12, C4.Dic3, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C2×C24, C3×SD16, C4○D12, C4○D12, S3×D4, S3×D4, D4⋊2S3, D4⋊2S3, S3×Q8, S3×Q8, Q8⋊3S3, Q8⋊3S3, C2×C3⋊D4, C6×D4, C6×Q8, D4○SD16, C8○D12, C4○D24, S3×SD16, Q8⋊3D6, D4.D6, Q8.7D6, D12⋊6C22, Q8.11D6, C6×SD16, D4⋊6D6, Q8.15D6, SD16⋊13D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, S3×D4, S3×C23, D4○SD16, C2×S3×D4, SD16⋊13D6

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E2F2G2H 3 4A4B4C4D4E4F4G4H6A6B6C6D6E8A8B8C8D8E12A12B12C12D24A24B24C24D
order12222222234444444466666888881212121224242424
size112446612122224466121222288224121244884444

36 irreducible representations

dim111111111111222222224444
type++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2S3D4D4D4D6D6D6D6S3×D4S3×D4D4○SD16SD16⋊13D6
kernelSD16⋊13D6C8○D12C4○D24S3×SD16Q8⋊3D6D4.D6Q8.7D6D12⋊6C22Q8.11D6C6×SD16D4⋊6D6Q8.15D6C2×SD16Dic6D12C3⋊D4C2×C8SD16C2×D4C2×Q8C4C22C3C1
# reps111222211111111214111124

Matrix representation of SD16⋊13D6 ►in GL8(𝔽73)

007100000
000710000
370000000
037000000
0000611200
000067000
00004360069
000056171861
,
720000000
072000000
00100000
00010000
000039343661
00005617180
00006181713
000066100
,
7272000000
10000000
00110000
007200000
00001000
000017200
00006810148
000000072
,
7272000000
01000000
0072720000
00010000
00001000
00000100
00001063720
000000072

G:=sub<GL(8,GF(73))| [0,0,37,0,0,0,0,0,0,0,0,37,0,0,0,0,71,0,0,0,0,0,0,0,0,71,0,0,0,0,0,0,0,0,0,0,61,67,43,56,0,0,0,0,12,0,60,17,0,0,0,0,0,0,0,18,0,0,0,0,0,0,69,61],[72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,39,56,61,6,0,0,0,0,34,17,8,61,0,0,0,0,36,18,17,0,0,0,0,0,61,0,13,0],[72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,68,0,0,0,0,0,0,72,10,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,48,72],[72,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,1,0,10,0,0,0,0,0,0,1,63,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72] >;
 

SD16⋊13D6 in GAP, Magma, Sage, TeX 

{\rm SD}_{16}\rtimes_{13}D_6
 
% in TeX
 
G:=Group("SD16:13D6");
 
// GroupNames label
 
G:=SmallGroup(192,1321);
 
// by ID
 
G=gap.SmallGroup(192,1321);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,185,136,438,235,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^6=d^2=1,b*a*b=c*a*c^-1=a^3,a*d=d*a,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
 
// generators/relations
 

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