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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 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×5], S3 [×2], C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×9], D4, D4 [×7], Q8, Q8 [×5], C23, Dic3 [×3], C12 [×2], C12 [×2], D6 [×4], C2×C6, C2×C6, C42, C22⋊C4 [×2], M4(2), M4(2), D8 [×2], SD16 [×2], C2×D4, C2×Q8 [×3], C4○D4, C4○D4 [×5], C3⋊C8, C24, Dic6, Dic6 [×4], C4×S3 [×3], D12, D12 [×2], C2×Dic3 [×2], C2×Dic3 [×2], C3⋊D4 [×3], C2×C12, C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C22×S3, C4.10D4, C4≀C2, C4≀C2, C4.4D4, C8⋊C22 [×2], 2- 1+4, C24⋊C2, D24, C4.Dic3, D6⋊C4 [×2], D4⋊S3, Q82S3, C4×C12, C3×M4(2), C2×Dic6, C2×Dic6, C2×D12, C4○D12, C4○D12, D42S3 [×3], 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 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], D12 [×2], C22×S3, C22≀C2, C2×D12, S3×D4 [×2], 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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(25 42)(26 41)(27 40)(28 39)(29 38)(30 37)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 34 19 28)(14 35 20 29)(15 36 21 30)(16 25 22 31)(17 26 23 32)(18 27 24 33)
(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 43 19 37)(14 44 20 38)(15 45 21 39)(16 46 22 40)(17 47 23 41)(18 48 24 42)

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

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

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

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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