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G = D12.40D4order 192 = 26·3

10th non-split extension by D12 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D12.40D4, Dic6.40D4, M4(2).17D6, C4○D4.24D6, (C3×D4).17D4, C4.107(S3×D4), C8.C223S3, Q8○D12.2C2, (C3×Q8).17D4, (C2×Q8).95D6, D12⋊C48C2, C6.66C22≀C2, C12.201(C2×D4), (C2×Dic3).6D4, C22.38(S3×D4), Q83Dic39C2, Dic3⋊Q87C2, Q8.11D66C2, C12.47D48C2, D4.12(C3⋊D4), C34(D4.10D4), (C2×C12).20C23, Q8.19(C3⋊D4), (C6×Q8).98C22, C2.34(C232D6), C4○D12.26C22, (C4×Dic3).59C22, C4.Dic3.29C22, (C2×Dic6).137C22, (C3×M4(2)).14C22, (C2×C6).37(C2×D4), C4.57(C2×C3⋊D4), (C3×C8.C22)⋊7C2, (C2×C4).20(C22×S3), (C3×C4○D4).18C22, SmallGroup(192,764)

Series: Derived Chief Lower central Upper central

C1C2×C12 — D12.40D4
C1C3C6C12C2×C12C4○D12Q8○D12 — D12.40D4
C3C6C2×C12 — D12.40D4
C1C2C2×C4C8.C22

Generators and relations for D12.40D4
 G = < a,b,c,d | a12=b2=1, c4=d2=a6, bab=a-1, cac-1=a7, ad=da, cbc-1=a3b, bd=db, dcd-1=a6c3 >

Subgroups: 400 in 142 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×7], C22, C22 [×2], S3, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×10], D4, D4 [×5], Q8, Q8 [×7], Dic3 [×5], C12 [×2], C12 [×2], D6, C2×C6, C2×C6, C42, C4⋊C4 [×2], M4(2), M4(2), SD16 [×2], Q16 [×2], C2×Q8, C2×Q8 [×3], C4○D4, C4○D4 [×5], C3⋊C8, C24, Dic6, Dic6 [×4], C4×S3 [×3], D12, C2×Dic3 [×2], C2×Dic3 [×3], C3⋊D4 [×3], C2×C12, C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C3×Q8 [×2], C4.10D4, C4≀C2 [×2], C4⋊Q8, C8.C22, C8.C22, 2- 1+4, C4.Dic3, C4×Dic3, Dic3⋊C4 [×2], Q82S3, C3⋊Q16, C3×M4(2), C3×SD16, C3×Q16, C2×Dic6, C2×Dic6, C4○D12, C4○D12, D42S3 [×3], S3×Q8, C6×Q8, C3×C4○D4, D4.10D4, C12.47D4, D12⋊C4, Q83Dic3, Q8.11D6, Dic3⋊Q8, C3×C8.C22, Q8○D12, D12.40D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C22≀C2, S3×D4 [×2], C2×C3⋊D4, D4.10D4, C232D6, D12.40D4

Character table of D12.40D4

 class 12A2B2C2D34A4B4C4D4E4F4G4H4I6A6B6C8A8B12A12B12C12D12E24A24B
 size 1124122224812121212122488244488888
ρ1111111111111111111111111111    trivial
ρ2111-1-1111-111-111111-1-1-11111-1-1-1    linear of order 2
ρ3111111111-1-1111-1111-1-111-1-11-1-1    linear of order 2
ρ4111-1-1111-1-1-1-111-111-11111-1-1-111    linear of order 2
ρ5111-11111-1-111-1-1111-11-111-1-1-111    linear of order 2
ρ61111-11111-11-1-1-11111-1111-1-11-1-1    linear of order 2
ρ71111-111111-1-1-1-1-11111-11111111    linear of order 2
ρ8111-11111-11-11-1-1-111-1-111111-1-1-1    linear of order 2
ρ922-2022-22000-20002-20002-200000    orthogonal lifted from D4
ρ1022-2-2022-220000002-2-200-2200200    orthogonal lifted from D4
ρ1122220-1222200000-1-1-120-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1222-20-22-2200020002-20002-200000    orthogonal lifted from D4
ρ13222-20-122-2200000-1-11-20-1-1-1-1111    orthogonal lifted from D6
ρ1422-22022-2-20000002-2200-2200-200    orthogonal lifted from D4
ρ15222002-2-20000-22022000-2-200000    orthogonal lifted from D4
ρ16222-20-122-2-200000-1-1120-1-1111-1-1    orthogonal lifted from D6
ρ1722220-1222-200000-1-1-1-20-1-111-111    orthogonal lifted from D6
ρ18222002-2-200002-2022000-2-200000    orthogonal lifted from D4
ρ1922-2-20-12-22000000-111001-1--3-3-1--3-3    complex lifted from C3⋊D4
ρ2022-220-12-2-2000000-11-1001-1--3-31-3--3    complex lifted from C3⋊D4
ρ2122-220-12-2-2000000-11-1001-1-3--31--3-3    complex lifted from C3⋊D4
ρ2222-2-20-12-22000000-111001-1-3--3-1-3--3    complex lifted from C3⋊D4
ρ2344-400-2-440000000-22000-2200000    orthogonal lifted from S3×D4
ρ2444400-2-4-40000000-2-20002200000    orthogonal lifted from S3×D4
ρ254-400040000-20002-400000000000    symplectic lifted from D4.10D4, Schur index 2
ρ264-4000400002000-2-400000000000    symplectic lifted from D4.10D4, Schur index 2
ρ278-8000-4000000000400000000000    symplectic faithful, Schur index 2

Smallest permutation representation of D12.40D4
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 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)
(1 46 10 37 7 40 4 43)(2 41 11 44 8 47 5 38)(3 48 12 39 9 42 6 45)(13 28 22 31 19 34 16 25)(14 35 23 26 20 29 17 32)(15 30 24 33 21 36 18 27)
(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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31), (1,46,10,37,7,40,4,43)(2,41,11,44,8,47,5,38)(3,48,12,39,9,42,6,45)(13,28,22,31,19,34,16,25)(14,35,23,26,20,29,17,32)(15,30,24,33,21,36,18,27), (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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31), (1,46,10,37,7,40,4,43)(2,41,11,44,8,47,5,38)(3,48,12,39,9,42,6,45)(13,28,22,31,19,34,16,25)(14,35,23,26,20,29,17,32)(15,30,24,33,21,36,18,27), (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)], [(1,48),(2,47),(3,46),(4,45),(5,44),(6,43),(7,42),(8,41),(9,40),(10,39),(11,38),(12,37),(13,30),(14,29),(15,28),(16,27),(17,26),(18,25),(19,36),(20,35),(21,34),(22,33),(23,32),(24,31)], [(1,46,10,37,7,40,4,43),(2,41,11,44,8,47,5,38),(3,48,12,39,9,42,6,45),(13,28,22,31,19,34,16,25),(14,35,23,26,20,29,17,32),(15,30,24,33,21,36,18,27)], [(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 D12.40D4 in GL8(𝔽73)

11000000
720000000
00110000
007200000
000007200
00001000
000034397270
0000260251
,
24496940000
2549840000
46949240000
656948240000
000000270
000042314665
000046000
000044472842
,
4925480000
482465690000
696524480000
8425490000
000042314665
000000460
000027000
00002926031
,
696524480000
8425490000
4925480000
482465690000
00000010
0000393413
000072000
00004528039

G:=sub<GL(8,GF(73))| [1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,34,26,0,0,0,0,72,0,39,0,0,0,0,0,0,0,72,25,0,0,0,0,0,0,70,1],[24,25,4,65,0,0,0,0,49,49,69,69,0,0,0,0,69,8,49,48,0,0,0,0,4,4,24,24,0,0,0,0,0,0,0,0,0,42,46,44,0,0,0,0,0,31,0,47,0,0,0,0,27,46,0,28,0,0,0,0,0,65,0,42],[49,48,69,8,0,0,0,0,25,24,65,4,0,0,0,0,4,65,24,25,0,0,0,0,8,69,48,49,0,0,0,0,0,0,0,0,42,0,27,29,0,0,0,0,31,0,0,26,0,0,0,0,46,46,0,0,0,0,0,0,65,0,0,31],[69,8,49,48,0,0,0,0,65,4,25,24,0,0,0,0,24,25,4,65,0,0,0,0,48,49,8,69,0,0,0,0,0,0,0,0,0,39,72,45,0,0,0,0,0,34,0,28,0,0,0,0,1,1,0,0,0,0,0,0,0,3,0,39] >;

D12.40D4 in GAP, Magma, Sage, TeX

D_{12}._{40}D_4
% in TeX

G:=Group("D12.40D4");
// GroupNames label

G:=SmallGroup(192,764);
// by ID

G=gap.SmallGroup(192,764);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,184,1123,297,136,851,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=1,c^4=d^2=a^6,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=a^6*c^3>;
// generators/relations

Export

Character table of D12.40D4 in TeX

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