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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, C8○D45S3, (C2×C8).80D6, C4○D4.47D6, (C3×D4).20D4, C12.42(C2×D4), C4.19(C2×D12), (C3×Q8).20D4, Q8.14D63C2, D4⋊D6.1C2, C34(D4.3D4), C8.44(C3⋊D4), C24.C414C2, C6.76(C4⋊D4), (C2×C24).66C22, C12.47D413C2, C12.46D413C2, C2.24(C127D4), (C2×C12).421C23, C22.8(C4○D12), (C2×D12).111C22, C4.Dic3.16C22, (C2×Dic6).117C22, (C3×M4(2)).36C22, (C3×C8○D4)⋊1C2, (C2×C24⋊C2)⋊3C2, (C2×C6).6(C4○D4), C4.117(C2×C3⋊D4), (C2×C4).123(C22×S3), (C3×C4○D4).36C22, SmallGroup(192,700)

Series: Derived Chief Lower central Upper central

C1C2×C12 — Q8.8D12
C1C3C6C12C2×C12C2×D12C2×C24⋊C2 — Q8.8D12
C3C6C2×C12 — Q8.8D12
C1C2C2×C4C8○D4

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 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×3], S3, C6, C6 [×2], C8 [×2], C8 [×3], C2×C4, C2×C4 [×2], D4, D4 [×3], Q8, Q8 [×2], C23, Dic3, C12 [×2], C12, D6 [×2], C2×C6, C2×C6, C2×C8, C2×C8, M4(2), M4(2) [×3], D8, SD16 [×4], Q16, C2×D4, C2×Q8, C4○D4, C3⋊C8 [×2], C24 [×2], C24, Dic6 [×2], D12 [×2], C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C24⋊C2 [×2], C4.Dic3 [×2], D4⋊S3, D4.S3, Q82S3, C3⋊Q16, C2×C24, C2×C24, C3×M4(2), C3×M4(2), C2×Dic6, C2×D12, C3×C4○D4, D4.3D4, C24.C4, C12.46D4, C12.47D4, C2×C24⋊C2, D4⋊D6, Q8.14D6, C3×C8○D4, Q8.8D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, D12 [×2], C3⋊D4 [×2], C22×S3, C4⋊D4, C2×D12, C4○D12, C2×C3⋊D4, D4.3D4, C127D4, 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 28 13 40)(2 29 14 41)(3 30 15 42)(4 31 16 43)(5 32 17 44)(6 33 18 45)(7 34 19 46)(8 35 20 47)(9 36 21 48)(10 37 22 25)(11 38 23 26)(12 39 24 27)
(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 36 37 48)(26 47 38 35)(27 34 39 46)(28 45 40 33)(29 32 41 44)(30 43 42 31)

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

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

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

36 conjugacy classes

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

36 irreducible representations

dim1111111122222222222244
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D4D6D6D6C4○D4C3⋊D4D12D12C4○D12D4.3D4Q8.8D12
kernelQ8.8D12C24.C4C12.46D4C12.47D4C2×C24⋊C2D4⋊D6Q8.14D6C3×C8○D4C8○D4C24C3×D4C3×Q8C2×C8M4(2)C4○D4C2×C6C8D4Q8C22C3C1
# 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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