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G = Q8.8D12order 192 = 26·3

3rd non-split extension by Q8 of D12 acting via D12/C12=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.3D12, C24.87D4, Q8.8D12, M4(2).33D6, C8oD4:5S3, (C2xC8).80D6, C4oD4.47D6, (C3xD4).20D4, C12.42(C2xD4), C4.19(C2xD12), (C3xQ8).20D4, Q8.14D6:3C2, D4:D6.1C2, C3:4(D4.3D4), C8.44(C3:D4), C24.C4:14C2, C6.76(C4:D4), (C2xC24).66C22, C12.47D4:13C2, C12.46D4:13C2, C2.24(C12:7D4), (C2xC12).421C23, C22.8(C4oD12), (C2xD12).111C22, C4.Dic3.16C22, (C2xDic6).117C22, (C3xM4(2)).36C22, (C3xC8oD4):1C2, (C2xC24:C2):3C2, (C2xC6).6(C4oD4), C4.117(C2xC3:D4), (C2xC4).123(C22xS3), (C3xC4oD4).36C22, SmallGroup(192,700)

Series: Derived Chief Lower central Upper central

C1C2xC12 — Q8.8D12
C1C3C6C12C2xC12C2xD12C2xC24:C2 — Q8.8D12
C3C6C2xC12 — Q8.8D12
C1C2C2xC4C8oD4

Generators and relations for Q8.8D12
 G = < a,b,c,d | a4=1, b2=c12=d2=a2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c11 >

Subgroups: 320 in 104 conjugacy classes, 39 normal (all 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, C2xC6, C2xC6, C2xC8, C2xC8, M4(2), M4(2), D8, SD16, Q16, C2xD4, C2xQ8, C4oD4, C3:C8, C24, C24, Dic6, D12, C2xDic3, C2xC12, C2xC12, C3xD4, C3xD4, C3xQ8, C22xS3, C4.D4, C4.10D4, C8.C4, C8oD4, C2xSD16, C8:C22, C8.C22, C24:C2, C4.Dic3, D4:S3, D4.S3, Q8:2S3, C3:Q16, C2xC24, C2xC24, C3xM4(2), C3xM4(2), C2xDic6, C2xD12, C3xC4oD4, D4.3D4, C24.C4, C12.46D4, C12.47D4, C2xC24:C2, D4:D6, Q8.14D6, C3xC8oD4, Q8.8D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, D12, C3:D4, C22xS3, C4:D4, C2xD12, C4oD12, C2xC3:D4, D4.3D4, C12:7D4, Q8.8D12

Smallest permutation representation of Q8.8D12
On 48 points
Generators in S48
(1 7 13 19)(2 8 14 20)(3 9 15 21)(4 10 16 22)(5 11 17 23)(6 12 18 24)(25 43 37 31)(26 44 38 32)(27 45 39 33)(28 46 40 34)(29 47 41 35)(30 48 42 36)
(1 43 13 31)(2 44 14 32)(3 45 15 33)(4 46 16 34)(5 47 17 35)(6 48 18 36)(7 25 19 37)(8 26 20 38)(9 27 21 39)(10 28 22 40)(11 29 23 41)(12 30 24 42)
(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 12 13 24)(2 23 14 11)(3 10 15 22)(4 21 16 9)(5 8 17 20)(6 19 18 7)(25 30 37 42)(26 41 38 29)(27 28 39 40)(31 48 43 36)(32 35 44 47)(33 46 45 34)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D6A6B6C6D8A8B8C8D8E8F8G12A12B12C12D12E24A24B24C24D24E···24J
order12222344446666888888812121212122424242424···24
size11242422242424442244424242244422224···4

36 irreducible representations

dim1111111122222222222244
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D4D6D6D6C4oD4C3:D4D12D12C4oD12D4.3D4Q8.8D12
kernelQ8.8D12C24.C4C12.46D4C12.47D4C2xC24:C2D4:D6Q8.14D6C3xC8oD4C8oD4C24C3xD4C3xQ8C2xC8M4(2)C4oD4C2xC6C8D4Q8C22C3C1
# reps1111111112111112422424

Matrix representation of Q8.8D12 in GL4(F73) generated by

665900
14700
460714
0465966
,
2705945
0272814
00460
00046
,
483700
361100
004837
003611
,
483700
622500
16163625
0576237
G:=sub<GL(4,GF(73))| [66,14,46,0,59,7,0,46,0,0,7,59,0,0,14,66],[27,0,0,0,0,27,0,0,59,28,46,0,45,14,0,46],[48,36,0,0,37,11,0,0,0,0,48,36,0,0,37,11],[48,62,16,0,37,25,16,57,0,0,36,62,0,0,25,37] >;

Q8.8D12 in GAP, Magma, Sage, TeX

Q_8._8D_{12}
% in TeX

G:=Group("Q8.8D12");
// GroupNames label

G:=SmallGroup(192,700);
// by ID

G=gap.SmallGroup(192,700);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,1123,297,136,1684,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=1,b^2=c^12=d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^11>;
// generators/relations

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