Copied to
clipboard

G = D12.40D4  order 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, C4oD4.24D6, (C3xD4).17D4, C4.107(S3xD4), C8.C22:3S3, Q8oD12.2C2, (C3xQ8).17D4, (C2xQ8).95D6, D12:C4:8C2, C6.66C22wrC2, C12.201(C2xD4), (C2xDic3).6D4, C22.38(S3xD4), Q8:3Dic3:9C2, Dic3:Q8:7C2, Q8.11D6:6C2, C12.47D4:8C2, D4.12(C3:D4), C3:4(D4.10D4), (C2xC12).20C23, Q8.19(C3:D4), (C6xQ8).98C22, C2.34(C23:2D6), C4oD12.26C22, (C4xDic3).59C22, C4.Dic3.29C22, (C2xDic6).137C22, (C3xM4(2)).14C22, (C2xC6).37(C2xD4), C4.57(C2xC3:D4), (C3xC8.C22):7C2, (C2xC4).20(C22xS3), (C3xC4oD4).18C22, SmallGroup(192,764)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2xC12 — D12.40D4
Chief series
C1 — C3 — C6 — C12 — C2xC12 — C4oD12 — Q8oD12 — D12.40D4
Lower central
C3 — C6 — C2xC12 — D12.40D4
Upper central
C1 — C2 — C2xC4 — C8.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, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, Dic3, C12, C12, D6, C2xC6, C2xC6, C42, C4:C4, M4(2), M4(2), SD16, Q16, C2xQ8, C2xQ8, C4oD4, C4oD4, C3:C8, C24, Dic6, Dic6, C4xS3, D12, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C3xD4, C3xQ8, C3xQ8, C4.10D4, C4wrC2, C4:Q8, C8.C22, C8.C22, 2- 1+4, C4.Dic3, C4xDic3, Dic3:C4, Q8:2S3, C3:Q16, C3xM4(2), C3xSD16, C3xQ16, C2xDic6, C2xDic6, C4oD12, C4oD12, D4:2S3, S3xQ8, C6xQ8, C3xC4oD4, D4.10D4, C12.47D4, D12:C4, Q8:3Dic3, Q8.11D6, Dic3:Q8, C3xC8.C22, Q8oD12, D12.40D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C3:D4, C22xS3, C22wrC2, S3xD4, C2xC3:D4, D4.10D4, C23:2D6, 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 S3xD4
ρ2444400-2-4-40000000-2-20002200000    orthogonal lifted from S3xD4
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 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)

(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 34 22 25 19 28 16 31)(14 29 23 32 20 35 17 26)(15 36 24 27 21 30 18 33)

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

׿
x
:
Z
F
o
wr
Q
<