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G = D8:6D6order 192 = 26·3

6th semidirect product of D8 and D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D8:6D6, SD16:4D6, D12.42D4, C24.3C23, M4(2):10D6, C12.22C24, Dic6.42D4, Dic12:2C22, D12.15C23, Dic6.15C23, C4oD4:7D6, D8:S3:4C2, C8:C22:5S3, Q8oD12:7C2, C3:D4.5D4, D8:3S3:2C2, D4:6D6:8C2, D12.C4:2C2, (S3xC8):4C22, (S3xSD16):2C2, D6.33(C2xD4), C3:4(D4oSD16), D4.D6:2C2, C4.116(S3xD4), C8.D6:2C2, (C3xD8):4C22, C3:C8.26C23, C8.3(C22xS3), (S3xQ8):3C22, C8:S3:4C22, C24:C2:4C22, D4:S3:15C22, Q8.13D6:4C2, (C2xD4).117D6, C12.243(C2xD4), C4.22(S3xC23), (S3xD4).3C22, C22.15(S3xD4), D4:2S3:4C22, (C4xS3).14C23, D4.S3:14C22, Dic3.38(C2xD4), (C3xSD16):4C22, C3:Q16:13C22, D4.15(C22xS3), (C3xD4).15C23, C6.123(C22xD4), Q8.25(C22xS3), (C3xQ8).15C23, (C2xC12).113C23, Q8:2S3:14C22, (C2xDic6):40C22, C4oD12.29C22, (C6xD4).168C22, (C3xM4(2)):4C22, C2.96(C2xS3xD4), (C3xC8:C22):4C2, (C2xC3:C8):18C22, (C2xC6).68(C2xD4), (C2xD4.S3):29C2, (C3xC4oD4):7C22, (C2xC4).97(C22xS3), SmallGroup(192,1334)

Series: Derived Chief Lower central Upper central

C1C12 — D8:6D6
C1C3C6C12C4xS3C4oD12D4:6D6 — D8:6D6
C3C6C12 — D8:6D6
C1C2C2xC4C8:C22

Generators and relations for D8:6D6
 G = < a,b,c,d | a8=b2=c6=d2=1, bab=a-1, cac-1=dad=a3, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 704 in 258 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2xC4, C2xC4, D4, D4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C2xC8, M4(2), M4(2), D8, D8, SD16, SD16, Q16, C2xD4, C2xD4, C2xQ8, C4oD4, C4oD4, C3:C8, C24, Dic6, Dic6, Dic6, C4xS3, C4xS3, D12, C2xDic3, C3:D4, C3:D4, C2xC12, C2xC12, C3xD4, C3xD4, C3xD4, C3xQ8, C22xS3, C22xC6, C8oD4, C2xSD16, C4oD8, C8:C22, C8:C22, C8.C22, 2+ 1+4, 2- 1+4, S3xC8, C8:S3, C24:C2, Dic12, C2xC3:C8, D4:S3, D4.S3, D4.S3, Q8:2S3, C3:Q16, C3xM4(2), C3xD8, C3xSD16, C2xDic6, C2xDic6, C4oD12, C4oD12, S3xD4, S3xD4, D4:2S3, D4:2S3, S3xQ8, C2xC3:D4, C6xD4, C3xC4oD4, D4oSD16, D12.C4, C8.D6, D8:S3, D8:3S3, S3xSD16, D4.D6, C2xD4.S3, Q8.13D6, C3xC8:C22, D4:6D6, Q8oD12, D8:6D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C24, C22xS3, C22xD4, S3xD4, S3xC23, D4oSD16, C2xS3xD4, D8:6D6

Smallest permutation representation of D8:6D6
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 19)(20 24)(21 23)(25 27)(28 32)(29 31)(33 35)(36 40)(37 39)(42 48)(43 47)(44 46)
(1 38 45 18 26 12)(2 33 46 21 27 15)(3 36 47 24 28 10)(4 39 48 19 29 13)(5 34 41 22 30 16)(6 37 42 17 31 11)(7 40 43 20 32 14)(8 35 44 23 25 9)
(1 43)(2 46)(3 41)(4 44)(5 47)(6 42)(7 45)(8 48)(9 19)(10 22)(11 17)(12 20)(13 23)(14 18)(15 21)(16 24)(25 29)(26 32)(28 30)(34 36)(35 39)(38 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), (2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(17,19)(20,24)(21,23)(25,27)(28,32)(29,31)(33,35)(36,40)(37,39)(42,48)(43,47)(44,46), (1,38,45,18,26,12)(2,33,46,21,27,15)(3,36,47,24,28,10)(4,39,48,19,29,13)(5,34,41,22,30,16)(6,37,42,17,31,11)(7,40,43,20,32,14)(8,35,44,23,25,9), (1,43)(2,46)(3,41)(4,44)(5,47)(6,42)(7,45)(8,48)(9,19)(10,22)(11,17)(12,20)(13,23)(14,18)(15,21)(16,24)(25,29)(26,32)(28,30)(34,36)(35,39)(38,40)>;

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

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

33 conjugacy classes

class 1 2A2B2C2D2E2F2G2H 3 4A4B4C4D4E4F4G4H6A6B6C6D6E8A8B8C8D8E12A12B12C24A24B
order12222222234444444466666888881212122424
size11244466122224661212122488844661244888

33 irreducible representations

dim1111111111112222222224448
type+++++++++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2S3D4D4D4D6D6D6D6D6S3xD4S3xD4D4oSD16D8:6D6
kernelD8:6D6D12.C4C8.D6D8:S3D8:3S3S3xSD16D4.D6C2xD4.S3Q8.13D6C3xC8:C22D4:6D6Q8oD12C8:C22Dic6D12C3:D4M4(2)D8SD16C2xD4C4oD4C4C22C3C1
# reps1112222111111112122111121

Matrix representation of D8:6D6 in GL6(F73)

7200000
0720000
00001261
00670120
00676667
00067667
,
7200000
0720000
001000
0017200
000010
0010072
,
010000
7210000
0010710
0000721
0000720
0001720
,
7210000
010000
0017100
0007200
0007201
0007210

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,67,67,0,0,0,0,0,6,67,0,0,12,12,6,6,0,0,61,0,67,67],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,1,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72],[0,72,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,71,72,72,72,0,0,0,1,0,0],[72,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,71,72,72,72,0,0,0,0,0,1,0,0,0,0,1,0] >;

D8:6D6 in GAP, Magma, Sage, TeX

D_8\rtimes_6D_6
% in TeX

G:=Group("D8:6D6");
// GroupNames label

G:=SmallGroup(192,1334);
// by ID

G=gap.SmallGroup(192,1334);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,570,185,136,438,235,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^6=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^3,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations

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