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G = D4.11D12order 192 = 26·3

1st non-split extension by D4 of D12 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.11D12, Q8.16D12, D2411C22, M4(2)⋊20D6, C24.10C23, C12.61C24, D12.24C23, Dic1210C22, Dic6.24C23, (C2×C8)⋊6D6, C8○D47S3, D4○D123C2, Q8○D123C2, C4○D2411C2, C8⋊D611C2, (C2×C24)⋊9C22, C4○D4.54D6, (C3×D4).23D4, C4.27(C2×D12), C31(D4○SD16), C12.73(C2×D4), (C3×Q8).23D4, C8.D611C2, C4○D121C22, C4.58(S3×C23), C8.55(C22×S3), C22.3(C2×D12), C6.28(C22×D4), C24⋊C211C22, C2.30(C22×D12), (C2×C12).515C23, (C2×Dic6)⋊35C22, (C2×D12).175C22, (C3×M4(2))⋊22C22, (C3×C8○D4)⋊3C2, (C2×C6).8(C2×D4), (C2×C24⋊C2)⋊6C2, (C2×C4).226(C22×S3), (C3×C4○D4).45C22, SmallGroup(192,1310)

Series: Derived Chief Lower central Upper central

C1C12 — D4.11D12
C1C3C6C12D12C2×D12D4○D12 — D4.11D12
C3C6C12 — D4.11D12
C1C2C4○D4C8○D4

Generators and relations for D4.11D12
 G = < a,b,c,d | a4=b2=1, c12=d2=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c11 >

Subgroups: 752 in 258 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2 [×7], C3, C4, C4 [×3], C4 [×4], C22 [×3], C22 [×7], S3 [×4], C6, C6 [×3], C8, C8 [×3], C2×C4 [×3], C2×C4 [×9], D4 [×3], D4 [×13], Q8, Q8 [×7], C23 [×3], Dic3 [×4], C12, C12 [×3], D6 [×7], C2×C6 [×3], C2×C8 [×3], M4(2) [×3], D8 [×3], SD16 [×10], Q16 [×3], C2×D4 [×6], C2×Q8 [×4], C4○D4, C4○D4 [×10], C24, C24 [×3], Dic6, Dic6 [×3], Dic6 [×3], C4×S3 [×6], D12, D12 [×3], D12 [×3], C2×Dic3 [×3], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C22×S3 [×3], C8○D4, C2×SD16 [×3], C4○D8 [×3], C8⋊C22 [×3], C8.C22 [×3], 2+ 1+4, 2- 1+4, C24⋊C2, C24⋊C2 [×9], D24 [×3], Dic12 [×3], C2×C24 [×3], C3×M4(2) [×3], C2×Dic6 [×3], C2×D12 [×3], C4○D12 [×6], S3×D4 [×3], D42S3 [×3], S3×Q8, Q83S3, C3×C4○D4, D4○SD16, C2×C24⋊C2 [×3], C4○D24 [×3], C8⋊D6 [×3], C8.D6 [×3], C3×C8○D4, D4○D12, Q8○D12, D4.11D12
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, D12 [×4], C22×S3 [×7], C22×D4, C2×D12 [×6], S3×C23, D4○SD16, C22×D12, D4.11D12

Smallest permutation representation of D4.11D12
On 48 points
Generators in S48
(1 46 13 34)(2 47 14 35)(3 48 15 36)(4 25 16 37)(5 26 17 38)(6 27 18 39)(7 28 19 40)(8 29 20 41)(9 30 21 42)(10 31 22 43)(11 32 23 44)(12 33 24 45)
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 25)(17 26)(18 27)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)
(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 42 37 30)(26 29 38 41)(27 40 39 28)(31 36 43 48)(32 47 44 35)(33 34 45 46)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G2H 3 4A4B4C4D4E4F4G4H6A6B6C6D8A8B8C8D8E12A12B12C12D12E24A24B24C24D24E···24J
order12222222234444444466668888812121212122424242424···24
size112221212121222222121212122444224442244422224···4

42 irreducible representations

dim111111112222222244
type++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D6D6D6D12D12D4○SD16D4.11D12
kernelD4.11D12C2×C24⋊C2C4○D24C8⋊D6C8.D6C3×C8○D4D4○D12Q8○D12C8○D4C3×D4C3×Q8C2×C8M4(2)C4○D4D4Q8C3C1
# reps133331111313316224

Matrix representation of D4.11D12 in GL6(𝔽73)

100000
010000
0063112272
0030444872
00703400
00022072
,
7200000
0720000
00106230
004329250
00370330
00071531
,
72720000
100000
0067697
0067673534
0000055
0000461
,
110000
0720000
006763866
00661239
00006118
00006912

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,63,30,70,0,0,0,11,44,3,2,0,0,22,48,40,20,0,0,72,72,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,10,43,3,0,0,0,62,29,70,71,0,0,3,25,33,53,0,0,0,0,0,1],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,67,67,0,0,0,0,6,67,0,0,0,0,9,35,0,4,0,0,7,34,55,61],[1,0,0,0,0,0,1,72,0,0,0,0,0,0,67,6,0,0,0,0,6,6,0,0,0,0,38,12,61,69,0,0,66,39,18,12] >;

D4.11D12 in GAP, Magma, Sage, TeX

D_4._{11}D_{12}
% in TeX

G:=Group("D4.11D12");
// GroupNames label

G:=SmallGroup(192,1310);
// by ID

G=gap.SmallGroup(192,1310);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,675,80,1684,102,6278]);
// Polycyclic

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

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