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G = D8:15D6  order 192 = 26·3

4th semidirect product of D8 and D6 acting through Inn(D8)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D8:15D6, Q16:13D6, SD16:11D6, D12.45D4, D24:19C22, C24.38C23, C12.16C24, Dic6.45D4, D12.11C23, C4oD8:4S3, C4oD4:4D6, (S3xD8):7C2, (C2xC8):13D6, C3:3(D4oD8), D4oD12:5C2, C8oD12:7C2, Q8:3D6:6C2, C3:D4.1D4, C3:C8.7C23, D4:D6:7C2, (C2xD24):23C2, (S3xC8):8C22, D4:S3:3C22, D6.29(C2xD4), C4.143(S3xD4), (S3xD4):2C22, D24:C2:7C2, C22.8(S3xD4), (C2xC24):12C22, C12.349(C2xD4), (C3xD8):13C22, (C4xS3).9C23, C8.16(C22xS3), C4.16(S3xC23), C8:S3:11C22, (C2xD12):34C22, Dic3.34(C2xD4), Q8:2S3:2C22, (C3xQ16):11C22, Q8:3S3:2C22, D4.10(C22xS3), (C3xD4).10C23, C6.117(C22xD4), (C3xQ8).10C23, Q8.20(C22xS3), (C2xC12).533C23, C4oD12.54C22, (C3xSD16):11C22, C4.Dic3:30C22, C2.90(C2xS3xD4), (C3xC4oD8):5C2, (C2xC6).13(C2xD4), (C3xC4oD4):3C22, (C2xC4).232(C22xS3), SmallGroup(192,1328)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — D8:15D6
Chief series
C1 — C3 — C6 — C12 — C4xS3 — C4oD12 — D4oD12 — D8:15D6
Lower central
C3 — C6 — C12 — D8:15D6
Upper central
C1 — C2 — C2xC4 — C4oD8

Generators and relations for D8:15D6
 G = < a,b,c,d | a8=b2=c6=d2=1, bab=dad=a-1, ac=ca, cbc-1=a4b, dbd=a2b, dcd=c-1 >

Subgroups: 856 in 268 conjugacy classes, 99 normal (31 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C2xC8, C2xC8, M4(2), D8, D8, SD16, SD16, Q16, C2xD4, C4oD4, C4oD4, C3:C8, C24, Dic6, C4xS3, C4xS3, D12, D12, D12, C3:D4, C3:D4, C2xC12, C2xC12, C3xD4, C3xD4, C3xQ8, C22xS3, C8oD4, C2xD8, C4oD8, C4oD8, C8:C22, 2+ 1+4, S3xC8, C8:S3, D24, C4.Dic3, D4:S3, Q8:2S3, C2xC24, C3xD8, C3xSD16, C3xQ16, C2xD12, C2xD12, C4oD12, C4oD12, S3xD4, S3xD4, Q8:3S3, C3xC4oD4, D4oD8, C8oD12, C2xD24, S3xD8, Q8:3D6, D24:C2, D4:D6, C3xC4oD8, D4oD12, D8:15D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C24, C22xS3, C22xD4, S3xD4, S3xC23, D4oD8, C2xS3xD4, D8:15D6

Smallest permutation representation of D8:15D6
►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)

(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(17 21)(18 20)(22 24)(25 29)(26 28)(30 32)(34 40)(35 39)(36 38)(41 47)(42 46)(43 45)

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

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

36 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J 3 4A4B4C4D4E4F6A6B6C6D8A8B8C8D8E12A12B12C12D12E24A24B24C24D
order122222222223444444666688888121212121224242424
size112446612121212222446624882241212224884444

36 irreducible representations

dim1111111112222222224444
type++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2S3D4D4D4D6D6D6D6D6S3xD4S3xD4D4oD8D8:15D6
kernelD8:15D6C8oD12C2xD24S3xD8Q8:3D6D24:C2D4:D6C3xC4oD8D4oD12C4oD8Dic6D12C3:D4C2xC8D8SD16Q16C4oD4C4C22C3C1
# reps1112422121112112121124

Matrix representation of D8:15D6 ►in GL4(F73) generated by

00320
00032
570320
057032
,
1000
0100
10720
01072
,
7665914
7145945
766667
7146659
,
00685
00105
343900
53900
G:=sub<GL(4,GF(73))| [0,0,57,0,0,0,0,57,32,0,32,0,0,32,0,32],[1,0,1,0,0,1,0,1,0,0,72,0,0,0,0,72],[7,7,7,7,66,14,66,14,59,59,66,66,14,45,7,59],[0,0,34,5,0,0,39,39,68,10,0,0,5,5,0,0] >;
 

D8:15D6 in GAP, Magma, Sage, TeX 

D_8\rtimes_{15}D_6
 
% in TeX
 
G:=Group("D8:15D6");
 
// GroupNames label
 
G:=SmallGroup(192,1328);
 
// by ID
 
G=gap.SmallGroup(192,1328);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,387,570,185,438,235,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^4*b,d*b*d=a^2*b,d*c*d=c^-1>;
 
// generators/relations
 

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