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G = D4.10D12order 192 = 26·3

5th non-split extension by D4 of D12 acting via D12/D6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.10D12, D12.35D4, Q8.15D12, C42.26D6, Dic6.35D4, M4(2).8D6, C4≀C24S3, C8⋊D69C2, Q8○D121C2, (C3×D4).5D4, C12.6(C2×D4), (C3×Q8).5D4, D4⋊D62C2, C4○D4.20D6, C4.12(C2×D12), C4.128(S3×D4), C6.30C22≀C2, C32(D4.8D4), (C2×Dic3).3D4, C424S310C2, C22.32(S3×D4), C427S311C2, C12.47D42C2, (C4×C12).53C22, C2.33(D6⋊D4), (C2×C12).267C23, C4○D12.16C22, (C2×D12).71C22, (C2×Dic6).77C22, (C3×M4(2)).5C22, C4.Dic3.11C22, (C3×C4≀C2)⋊4C2, (C2×C6).29(C2×D4), (C3×C4○D4).8C22, (C2×C4).112(C22×S3), SmallGroup(192,386)

Series: Derived Chief Lower central Upper central

C1C2×C12 — D4.10D12
C1C3C6C12C2×C12C4○D12Q8○D12 — D4.10D12
C3C6C2×C12 — D4.10D12
C1C2C2×C4C4≀C2

Generators and relations for D4.10D12
 G = < a,b,c,d | a12=b2=c4=1, d2=a6, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=a9c-1 >

Subgroups: 480 in 146 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, M4(2), M4(2), D8, SD16, C2×D4, C2×Q8, C4○D4, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C4.10D4, C4≀C2, C4≀C2, C4.4D4, C8⋊C22, 2- 1+4, C24⋊C2, D24, C4.Dic3, D6⋊C4, D4⋊S3, Q82S3, C4×C12, C3×M4(2), C2×Dic6, C2×Dic6, C2×D12, C4○D12, C4○D12, D42S3, S3×Q8, C3×C4○D4, D4.8D4, C424S3, C12.47D4, C3×C4≀C2, C427S3, C8⋊D6, D4⋊D6, Q8○D12, D4.10D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C22≀C2, C2×D12, S3×D4, D4.8D4, D6⋊D4, D4.10D12

Character table of D4.10D12

 class 12A2B2C2D2E34A4B4C4D4E4F4G4H6A6B6C8A8B12A12B12C12D12E12F12G12H24A24B
 size 112412242224441212122488242244444888
ρ1111111111111111111111111111111    trivial
ρ2111-1-11111-1111-1111-1-1-11111111-1-1-1    linear of order 2
ρ3111-11-1111-111-11-111-1-111111111-1-1-1    linear of order 2
ρ41111-1-1111111-1-1-11111-11111111111    linear of order 2
ρ5111-1-1-1111-1-1-11-1111-11111-11-1-1-1-111    linear of order 2
ρ611111-11111-1-1111111-1-111-11-1-1-11-1-1    linear of order 2
ρ71111-111111-1-1-1-1-1111-1111-11-1-1-11-1-1    linear of order 2
ρ8111-111111-1-1-1-11-111-11-111-11-1-1-1-111    linear of order 2
ρ9222-200-122-2-2-2000-1-1120-1-11-11111-1-1    orthogonal lifted from D6
ρ10222-200-122-222000-1-11-20-1-1-1-1-1-1-1111    orthogonal lifted from D6
ρ112220002-2-200020-222000-2-20-2000000    orthogonal lifted from D4
ρ1222-22002-22-2000002-2200-2-202000-200    orthogonal lifted from D4
ρ1322-2-2002-222000002-2-200-2-202000200    orthogonal lifted from D4
ρ1422-202022-20000-202-2000220-2000000    orthogonal lifted from D4
ρ152220002-2-2000-20222000-2-20-2000000    orthogonal lifted from D4
ρ16222200-1222-2-2000-1-1-1-20-1-11-1111-111    orthogonal lifted from D6
ρ17222200-122222000-1-1-120-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1822-20-2022-20000202-2000220-2000000    orthogonal lifted from D4
ρ1922-2200-1-22-200000-11-10011-3-13-331-33    orthogonal lifted from D12
ρ2022-2-200-1-22200000-1110011-3-13-33-13-3    orthogonal lifted from D12
ρ2122-2200-1-22-200000-11-100113-1-33-313-3    orthogonal lifted from D12
ρ2222-2-200-1-22200000-11100113-1-33-3-1-33    orthogonal lifted from D12
ρ23444000-2-4-4000000-2-20002202000000    orthogonal lifted from S3×D4
ρ2444-4000-24-4000000-22000-2-202000000    orthogonal lifted from S3×D4
ρ254-400004000-2i2i000-4000000-2i02i2i-2i000    complex lifted from D4.8D4
ρ264-4000040002i-2i000-40000002i0-2i-2i2i000    complex lifted from D4.8D4
ρ274-40000-2000-2i2i00020000-2323ζ4+2ζ32+10ζ43+2ζ32+1ζ43+2ζ3+1ζ4+2ζ3+1000    complex faithful
ρ284-40000-20002i-2i00020000-2323ζ43+2ζ3+10ζ4+2ζ3+1ζ4+2ζ32+1ζ43+2ζ32+1000    complex faithful
ρ294-40000-2000-2i2i0002000023-23ζ4+2ζ3+10ζ43+2ζ3+1ζ43+2ζ32+1ζ4+2ζ32+1000    complex faithful
ρ304-40000-20002i-2i0002000023-23ζ43+2ζ32+10ζ4+2ζ32+1ζ4+2ζ3+1ζ43+2ζ3+1000    complex faithful

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

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

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

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

Matrix representation of D4.10D12 in GL6(𝔽73)

63520000
47110000
00727201
00217272
000001
0000720
,
6310000
47100000
0072000
000010
000100
00717211
,
72710000
110000
00272700
0019462727
0000270
0000027
,
100000
010000
004646270
0000460
0004600
0019462727

G:=sub<GL(6,GF(73))| [63,47,0,0,0,0,52,11,0,0,0,0,0,0,72,2,0,0,0,0,72,1,0,0,0,0,0,72,0,72,0,0,1,72,1,0],[63,47,0,0,0,0,1,10,0,0,0,0,0,0,72,0,0,71,0,0,0,0,1,72,0,0,0,1,0,1,0,0,0,0,0,1],[72,1,0,0,0,0,71,1,0,0,0,0,0,0,27,19,0,0,0,0,27,46,0,0,0,0,0,27,27,0,0,0,0,27,0,27],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,0,19,0,0,46,0,46,46,0,0,27,46,0,27,0,0,0,0,0,27] >;

D4.10D12 in GAP, Magma, Sage, TeX

D_4._{10}D_{12}
% in TeX

G:=Group("D4.10D12");
// GroupNames label

G:=SmallGroup(192,386);
// by ID

G=gap.SmallGroup(192,386);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,226,1123,136,851,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^4=1,d^2=a^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=a^9*c^-1>;
// generators/relations

Export

Character table of D4.10D12 in TeX

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