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G = D8⋊D6  order 192 = 26·3

2nd semidirect product of D8 and D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D8⋊2D6, C16⋊2D6, D16⋊2S3, D6.6D8, C48⋊4C22, Dic3.8D8, C24.14C23, D24.1C22, Dic12⋊4C22, C3⋊C8.2D4, (S3×D8)⋊4C2, C4.2(S3×D4), (C3×D16)⋊4C2, C3⋊D16⋊2C2, C3⋊C16⋊1C22, D8.S3⋊1C2, D6.C8⋊3C2, (C4×S3).7D4, C48⋊C2⋊3C2, C2.17(S3×D8), C12.8(C2×D4), C6.33(C2×D8), D8⋊3S3⋊3C2, C3⋊2(C16⋊C22), (C3×D8)⋊6C22, (S3×C8).3C22, C8.20(C22×S3), SmallGroup(192,470)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C24 — D8⋊D6
Chief series
C1 — C3 — C6 — C12 — C24 — S3×C8 — S3×D8 — D8⋊D6
Lower central
C3 — C6 — C12 — C24 — D8⋊D6
Upper central
C1 — C2 — C4 — C8 — D16

Generators and relations for D8⋊D6
 G = < a,b,c,d | a8=b2=c6=d2=1, bab=cac-1=dad=a-1, cbc-1=a5b, dbd=ab, dcd=c-1 >

Subgroups: 380 in 90 conjugacy classes, 31 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, Q8, C23, Dic3, Dic3, C12, D6, D6, C2×C6, C16, C16, C2×C8, D8, D8, SD16, Q16, C2×D4, C4○D4, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C22×S3, M5(2), D16, D16, SD32, C2×D8, C4○D8, C3⋊C16, C48, S3×C8, D24, Dic12, D4⋊S3, D4.S3, C3×D8, S3×D4, D4⋊2S3, C16⋊C22, D6.C8, C48⋊C2, C3⋊D16, D8.S3, C3×D16, S3×D8, D8⋊3S3, D8⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C22×S3, C2×D8, S3×D4, C16⋊C22, S3×D8, D8⋊D6

Character table of D8⋊D6

 class 12A2B2C2D2E34A4B4C6A6B6C8A8B8C1216A16B16C16D24A24B48A48B48C48D
 size 1168824226242161622124441212444444
ρ1111111111111111111111111111    trivial
ρ21111-11111-111-11111-1-1-1-111-1-1-1-1    linear of order 2
ρ311-1-11111-1-11-1111-11-1-11111-1-1-1-1    linear of order 2
ρ411-1-1-1111-111-1-111-1111-1-1111111    linear of order 2
ρ511-111-111-1-111111-1111-1-1111111    linear of order 2
ρ611-11-1-111-1111-111-11-1-11111-1-1-1-1    linear of order 2
ρ7111-11-111111-111111-1-1-1-111-1-1-1-1    linear of order 2
ρ8111-1-1-1111-11-1-111111111111111    linear of order 2
ρ9220220-1200-1-1-1220-12200-1-1-1-1-1-1    orthogonal lifted from S3
ρ10220-2-20-1200-111220-12200-1-1-1-1-1-1    orthogonal lifted from D6
ρ1122-200022-20200-2-2220000-2-20000    orthogonal lifted from D4
ρ122220002220200-2-2-220000-2-20000    orthogonal lifted from D4
ρ13220-220-1200-11-1220-1-2-200-1-11111    orthogonal lifted from D6
ρ142202-20-1200-1-11220-1-2-200-1-11111    orthogonal lifted from D6
ρ1522-20002-220200000-2√2-√2-√2√200√2-√2√2-√2    orthogonal lifted from D8
ρ162220002-2-20200000-2√2-√2√2-√200√2-√2√2-√2    orthogonal lifted from D8
ρ172220002-2-20200000-2-√2√2-√2√200-√2√2-√2√2    orthogonal lifted from D8
ρ1822-20002-220200000-2-√2√2√2-√200-√2√2-√2√2    orthogonal lifted from D8
ρ19440000-2400-200-4-40-20000220000    orthogonal lifted from S3×D4
ρ204-400004000-400-2√22√2000000-2√22√20000    orthogonal lifted from C16⋊C22
ρ214-400004000-4002√2-2√20000002√2-2√20000    orthogonal lifted from C16⋊C22
ρ22440000-2-400-20000022√2-2√20000-√2√2-√2√2    orthogonal lifted from S3×D8
ρ23440000-2-400-2000002-2√22√20000√2-√2√2-√2    orthogonal lifted from S3×D8
ρ244-40000-20002002√2-2√2000000-√2√22ζ167ζ3+ζ167-2ζ16ζ3-ζ16-2ζ1613ζ32-ζ1613+2ζ1611ζ32+ζ1611-2ζ167ζ3-ζ167+2ζ16ζ3+ζ16-2ζ165ζ32-ζ165+2ζ163ζ32+ζ163    complex faithful
ρ254-40000-2000200-2√22√2000000√2-√2-2ζ1613ζ32-ζ1613+2ζ1611ζ32+ζ1611-2ζ167ζ3-ζ167+2ζ16ζ3+ζ16-2ζ165ζ32-ζ165+2ζ163ζ32+ζ1632ζ167ζ3+ζ167-2ζ16ζ3-ζ16    complex faithful
ρ264-40000-20002002√2-2√2000000-√2√2-2ζ167ζ3-ζ167+2ζ16ζ3+ζ16-2ζ165ζ32-ζ165+2ζ163ζ32+ζ1632ζ167ζ3+ζ167-2ζ16ζ3-ζ16-2ζ1613ζ32-ζ1613+2ζ1611ζ32+ζ1611    complex faithful
ρ274-40000-2000200-2√22√2000000√2-√2-2ζ165ζ32-ζ165+2ζ163ζ32+ζ1632ζ167ζ3+ζ167-2ζ16ζ3-ζ16-2ζ1613ζ32-ζ1613+2ζ1611ζ32+ζ1611-2ζ167ζ3-ζ167+2ζ16ζ3+ζ16    complex faithful

Smallest permutation representation of D8⋊D6
►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 10)(2 9)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(17 31)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(24 32)(33 44)(34 43)(35 42)(36 41)(37 48)(38 47)(39 46)(40 45)

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

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

Matrix representation of D8⋊D6 ►in GL4(𝔽97) generated by

70900
07090
7070
0707
,
96951530
216782
153012
67829596
,
969600
1000
0011
00960
,
969600
0100
0011
00096
G:=sub<GL(4,GF(97))| [7,0,7,0,0,7,0,7,90,0,7,0,0,90,0,7],[96,2,15,67,95,1,30,82,15,67,1,95,30,82,2,96],[96,1,0,0,96,0,0,0,0,0,1,96,0,0,1,0],[96,0,0,0,96,1,0,0,0,0,1,0,0,0,1,96] >;
 

D8⋊D6 in GAP, Magma, Sage, TeX 

D_8\rtimes D_6
 
% in TeX
 
G:=Group("D8:D6");
 
// GroupNames label
 
G:=SmallGroup(192,470);
 
// by ID
 
G=gap.SmallGroup(192,470);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,135,346,185,192,851,438,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^6=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^5*b,d*b*d=a*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of D8⋊D6 in TeX

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