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G = D4.12D12order 192 = 26·3

2nd non-split extension by D4 of D12 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.12D12, Q8.17D12, D2412C22, M4(2)⋊21D6, C12.62C24, C24.11C23, D12.25C23, Dic1221C22, Dic6.25C23, (C2×C8)⋊7D6, C8○D48S3, C31(D4○D8), D4○D124C2, C4○D2412C2, (C2×D24)⋊15C2, C8⋊D612C2, C4○D4.55D6, (C3×D4).24D4, C12.74(C2×D4), C4.28(C2×D12), (C3×Q8).24D4, (C2×C24)⋊10C22, C4○D122C22, C8.53(C22×S3), C4.59(S3×C23), C6.29(C22×D4), C22.4(C2×D12), C24⋊C212C22, (C2×D12)⋊31C22, C2.31(C22×D12), (C2×C12).516C23, (C3×M4(2))⋊23C22, (C3×C8○D4)⋊4C2, (C2×C6).9(C2×D4), (C2×C4).227(C22×S3), (C3×C4○D4).46C22, SmallGroup(192,1311)

Series: Derived Chief Lower central Upper central

C1C12 — D4.12D12
C1C3C6C12D12C2×D12D4○D12 — D4.12D12
C3C6C12 — D4.12D12
C1C2C4○D4C8○D4

Generators and relations for D4.12D12
 G = < a,b,c,d | a4=b2=d2=1, c12=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c11 >

Subgroups: 880 in 268 conjugacy classes, 107 normal (16 characteristic)
C1, C2, C2 [×9], C3, C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], S3 [×6], C6, C6 [×3], C8, C8 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×3], D4 [×18], Q8, Q8 [×2], C23 [×6], Dic3 [×2], C12, C12 [×3], D6 [×12], C2×C6 [×3], C2×C8 [×3], M4(2) [×3], D8 [×9], SD16 [×6], Q16, C2×D4 [×12], C4○D4, C4○D4 [×8], C24, C24 [×3], Dic6 [×2], C4×S3 [×6], D12 [×6], D12 [×6], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C22×S3 [×6], C8○D4, C2×D8 [×3], C4○D8 [×3], C8⋊C22 [×6], 2+ 1+4 [×2], C24⋊C2 [×6], D24 [×9], Dic12, C2×C24 [×3], C3×M4(2) [×3], C2×D12 [×6], C4○D12 [×6], S3×D4 [×6], Q83S3 [×2], C3×C4○D4, D4○D8, C2×D24 [×3], C4○D24 [×3], C8⋊D6 [×6], C3×C8○D4, D4○D12 [×2], D4.12D12
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○D8, C22×D12, D4.12D12

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

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E···2J 3 4A4B4C4D4E4F6A6B6C6D8A8B8C8D8E12A12B12C12D12E24A24B24C24D24E···24J
order122222···2344444466668888812121212122424242424···24
size1122212···122222212122444224442244422224···4

42 irreducible representations

dim1111112222222244
type++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D6D12D12D4○D8D4.12D12
kernelD4.12D12C2×D24C4○D24C8⋊D6C3×C8○D4D4○D12C8○D4C3×D4C3×Q8C2×C8M4(2)C4○D4D4Q8C3C1
# reps1336121313316224

Matrix representation of D4.12D12 in GL4(𝔽73) generated by

00720
00072
1000
0100
,
1000
0100
00720
00072
,
685500
185000
006855
001850
,
501800
682300
005018
006823
G:=sub<GL(4,GF(73))| [0,0,1,0,0,0,0,1,72,0,0,0,0,72,0,0],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[68,18,0,0,55,50,0,0,0,0,68,18,0,0,55,50],[50,68,0,0,18,23,0,0,0,0,50,68,0,0,18,23] >;

D4.12D12 in GAP, Magma, Sage, TeX

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

G:=Group("D4.12D12");
// GroupNames label

G:=SmallGroup(192,1311);
// by ID

G=gap.SmallGroup(192,1311);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,675,192,1684,102,6278]);
// Polycyclic

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

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