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G = D8⋊13D6  order 192 = 26·3

2nd semidirect product of D8 and D6 acting through Inn(D8)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D8⋊13D6, D12.28D4, C12.4C24, D24⋊16C22, C24.40C23, Dic6.28D4, D12.2C23, Dic6.2C23, Dic12⋊14C22, (C2×C8)⋊9D6, (C6×D8)⋊3C2, (S3×D8)⋊6C2, C3⋊2(D4○D8), C4○D24⋊3C2, C8○D12⋊2C2, (C2×D4)⋊14D6, (C2×D8)⋊12S3, D8⋊S3⋊5C2, C4.75(S3×D4), C3⋊D4.8D4, C3⋊C8.1C23, D4⋊6D6⋊5C2, D8⋊3S3⋊6C2, (S3×C8)⋊7C22, (C2×C24)⋊3C22, D4⋊S3⋊1C22, D6.26(C2×D4), C12.79(C2×D4), (S3×D4)⋊1C22, C4.4(S3×C23), D12⋊6C22⋊7C2, C4○D12⋊3C22, (C3×D8)⋊11C22, (C6×D4)⋊20C22, (C4×S3).2C23, C8.10(C22×S3), D4.S3⋊1C22, D4.2(C22×S3), (C3×D4).2C23, C22.20(S3×D4), C24⋊C2⋊14C22, C8⋊S3⋊13C22, D4⋊2S3⋊1C22, Dic3.31(C2×D4), C6.105(C22×D4), (C2×C12).521C23, C4.Dic3⋊28C22, C2.78(C2×S3×D4), (C2×C6).394(C2×D4), (C2×C4).229(C22×S3), SmallGroup(192,1316)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — D8⋊13D6
Chief series
C1 — C3 — C6 — C12 — C4×S3 — C4○D12 — D4⋊6D6 — D8⋊13D6
Lower central
C3 — C6 — C12 — D8⋊13D6

Subgroups: 808 in 268 conjugacy classes, 99 normal (29 characteristic)
C1, C2, C2 [×9], C3, C4 [×2], C4 [×4], C22, C22 [×14], S3 [×4], C6, C6 [×5], C8 [×2], C8 [×2], C2×C4, C2×C4 [×8], D4 [×4], D4 [×17], Q8 [×3], C23 [×6], Dic3 [×2], Dic3 [×2], C12 [×2], D6 [×2], D6 [×6], C2×C6, C2×C6 [×6], C2×C8, C2×C8 [×2], M4(2) [×3], D8 [×4], D8 [×5], SD16 [×6], Q16, C2×D4 [×2], C2×D4 [×10], C4○D4 [×9], C3⋊C8 [×2], C24 [×2], Dic6, Dic6 [×2], C4×S3 [×2], C4×S3 [×2], D12, D12 [×2], C2×Dic3 [×4], C3⋊D4 [×2], C3⋊D4 [×10], C2×C12, C3×D4 [×4], C3×D4 [×2], C22×S3 [×4], C22×C6 [×2], C8○D4, C2×D8, C2×D8 [×2], C4○D8 [×3], C8⋊C22 [×6], 2+ (1+4) [×2], S3×C8 [×2], C8⋊S3 [×2], C24⋊C2 [×2], D24, Dic12, C4.Dic3, D4⋊S3 [×4], D4.S3 [×4], C2×C24, C3×D8 [×4], C4○D12, C4○D12 [×2], S3×D4 [×4], S3×D4 [×2], D4⋊2S3 [×4], D4⋊2S3 [×2], C2×C3⋊D4 [×4], C6×D4 [×2], D4○D8, C8○D12, C4○D24, S3×D8 [×2], D8⋊S3 [×4], D8⋊3S3 [×2], D12⋊6C22 [×2], C6×D8, D4⋊6D6 [×2], D8⋊13D6

Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C22×S3 [×7], C22×D4, S3×D4 [×2], S3×C23, D4○D8, C2×S3×D4, D8⋊13D6

Generators and relations
 G = < a,b,c,d | a8=b2=c6=d2=1, bab=cac-1=a-1, ad=da, cbc-1=a6b, dbd=a4b, dcd=c-1 >

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

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

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

Matrix representation ►G ⊆ GL4(𝔽7) generated by

3522
4512
6652
1630
,
3263
6045
6652
0006
,
0460
2520
1635
3636
,
1024
2255
6316
4263
G:=sub<GL(4,GF(7))| [3,4,6,1,5,5,6,6,2,1,5,3,2,2,2,0],[3,6,6,0,2,0,6,0,6,4,5,0,3,5,2,6],[0,2,1,3,4,5,6,6,6,2,3,3,0,0,5,6],[1,2,6,4,0,2,3,2,2,5,1,6,4,5,6,3] >;
 

36 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J 3 4A4B4C4D4E4F6A6B6C6D6E6F6G8A8B8C8D8E12A12B24A24B24C24D
order122222222223444444666666688888121224242424
size112444466121222266121222288882241212444444

36 irreducible representations

dim11111111122222224444
type+++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2S3D4D4D4D6D6D6S3×D4S3×D4D4○D8D8⋊13D6
kernelD8⋊13D6C8○D12C4○D24S3×D8D8⋊S3D8⋊3S3D12⋊6C22C6×D8D4⋊6D6C2×D8Dic6D12C3⋊D4C2×C8D8C2×D4C4C22C3C1
# reps11124221211121421124

In GAP, Magma, Sage, TeX 

D_8\rtimes_{13}D_6
 
% in TeX
 
G:=Group("D8:13D6");
 
// GroupNames label
 
G:=SmallGroup(192,1316);
 
// by ID
 
G=gap.SmallGroup(192,1316);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,185,438,235,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^6=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^6*b,d*b*d=a^4*b,d*c*d=c^-1>;
 
// generators/relations
 

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