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G = D8:D6order 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, (S3xD8):4C2, C4.2(S3xD4), (C3xD16):4C2, C3:D16:2C2, C3:C16:1C22, D8.S3:1C2, D6.C8:3C2, (C4xS3).7D4, C48:C2:3C2, C2.17(S3xD8), C12.8(C2xD4), C6.33(C2xD8), D8:3S3:3C2, C3:2(C16:C22), (C3xD8):6C22, (S3xC8).3C22, C8.20(C22xS3), SmallGroup(192,470)

Series: Derived Chief Lower central Upper central

C1C24 — D8:D6
C1C3C6C12C24S3xC8S3xD8 — D8:D6
C3C6C12C24 — D8:D6
C1C2C4C8D16

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, C2xC4, D4, Q8, C23, Dic3, Dic3, C12, D6, D6, C2xC6, C16, C16, C2xC8, D8, D8, SD16, Q16, C2xD4, C4oD4, C3:C8, C24, Dic6, C4xS3, D12, C2xDic3, C3:D4, C3xD4, C22xS3, M5(2), D16, D16, SD32, C2xD8, C4oD8, C3:C16, C48, S3xC8, D24, Dic12, D4:S3, D4.S3, C3xD8, S3xD4, D4:2S3, C16:C22, D6.C8, C48:C2, C3:D16, D8.S3, C3xD16, S3xD8, D8:3S3, D8:D6
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2xD4, C22xS3, C2xD8, S3xD4, C16:C22, S3xD8, 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
ρ1522-20002-220200000-22-2-22002-22-2    orthogonal lifted from D8
ρ162220002-2-20200000-22-22-2002-22-2    orthogonal lifted from D8
ρ172220002-2-20200000-2-22-2200-22-22    orthogonal lifted from D8
ρ1822-20002-220200000-2-222-200-22-22    orthogonal lifted from D8
ρ19440000-2400-200-4-40-20000220000    orthogonal lifted from S3xD4
ρ204-400004000-400-2222000000-22220000    orthogonal lifted from C16:C22
ρ214-400004000-40022-2200000022-220000    orthogonal lifted from C16:C22
ρ22440000-2-400-200000222-220000-22-22    orthogonal lifted from S3xD8
ρ23440000-2-400-2000002-222200002-22-2    orthogonal lifted from S3xD8
ρ244-40000-200020022-22000000-22167ζ3167-2ζ16ζ316-2ζ1613ζ321613+2ζ1611ζ321611-2ζ167ζ3167+2ζ16ζ316-2ζ165ζ32165+2ζ163ζ32163    complex faithful
ρ254-40000-2000200-22220000002-2-2ζ1613ζ321613+2ζ1611ζ321611-2ζ167ζ3167+2ζ16ζ316-2ζ165ζ32165+2ζ163ζ32163167ζ3167-2ζ16ζ316    complex faithful
ρ264-40000-200020022-22000000-22-2ζ167ζ3167+2ζ16ζ316-2ζ165ζ32165+2ζ163ζ32163167ζ3167-2ζ16ζ316-2ζ1613ζ321613+2ζ1611ζ321611    complex faithful
ρ274-40000-2000200-22220000002-2-2ζ165ζ32165+2ζ163ζ32163167ζ3167-2ζ16ζ316-2ζ1613ζ321613+2ζ1611ζ321611-2ζ167ζ3167+2ζ16ζ316    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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