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G = Q8.9D12order 192 = 26·3

4th non-split extension by Q8 of D12 acting via D12/C12=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.4D12, C24.83D4, Q8.9D12, M4(2).34D6, C8○D46S3, (C2×C8).81D6, D4⋊D63C2, (C2×D24)⋊13C2, C4○D4.48D6, (C3×D4).21D4, C4.20(C2×D12), C12.43(C2×D4), (C3×Q8).21D4, C33(D4.4D4), C8.40(C3⋊D4), C24.C415C2, C6.77(C4⋊D4), (C2×C24).67C22, C12.46D414C2, C2.25(C127D4), (C2×C12).422C23, C22.9(C4○D12), (C2×D12).112C22, C4.Dic3.17C22, (C3×M4(2)).37C22, (C3×C8○D4)⋊2C2, (C2×C6).7(C4○D4), C4.118(C2×C3⋊D4), (C2×C4).124(C22×S3), (C3×C4○D4).37C22, SmallGroup(192,701)

Series: Derived Chief Lower central Upper central

C1C2×C12 — Q8.9D12
C1C3C6C12C2×C12C2×D12C2×D24 — Q8.9D12
C3C6C2×C12 — Q8.9D12
C1C2C2×C4C8○D4

Generators and relations for Q8.9D12
 G = < a,b,c,d | a4=d2=1, b2=c12=a2, bab-1=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=a2c11 >

Subgroups: 384 in 108 conjugacy classes, 39 normal (31 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C12, C12, D6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, C2×D4, C4○D4, C3⋊C8, C24, C24, D12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, D24, C4.Dic3, D4⋊S3, Q82S3, C2×C24, C2×C24, C3×M4(2), C3×M4(2), C2×D12, C3×C4○D4, D4.4D4, C24.C4, C12.46D4, C2×D24, D4⋊D6, C3×C8○D4, Q8.9D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C3⋊D4, C22×S3, C4⋊D4, C2×D12, C4○D12, C2×C3⋊D4, D4.4D4, C127D4, Q8.9D12

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

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C6A6B6C6D8A8B8C8D8E8F8G12A12B12C12D12E24A24B24C24D24E···24J
order12222234446666888888812121212122424242424···24
size11242424222424442244424242244422224···4

36 irreducible representations

dim11111122222222222244
type+++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D4D6D6D6C4○D4C3⋊D4D12D12C4○D12D4.4D4Q8.9D12
kernelQ8.9D12C24.C4C12.46D4C2×D24D4⋊D6C3×C8○D4C8○D4C24C3×D4C3×Q8C2×C8M4(2)C4○D4C2×C6C8D4Q8C22C3C1
# reps11212112111112422424

Matrix representation of Q8.9D12 in GL4(𝔽73) generated by

71400
596600
006659
00147
,
006659
00147
665900
14700
,
231800
55500
002318
00555
,
231800
685000
005550
006818
G:=sub<GL(4,GF(73))| [7,59,0,0,14,66,0,0,0,0,66,14,0,0,59,7],[0,0,66,14,0,0,59,7,66,14,0,0,59,7,0,0],[23,55,0,0,18,5,0,0,0,0,23,55,0,0,18,5],[23,68,0,0,18,50,0,0,0,0,55,68,0,0,50,18] >;

Q8.9D12 in GAP, Magma, Sage, TeX

Q_8._9D_{12}
% in TeX

G:=Group("Q8.9D12");
// GroupNames label

G:=SmallGroup(192,701);
// by ID

G=gap.SmallGroup(192,701);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,1123,297,136,1684,102,6278]);
// Polycyclic

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

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