Copied to
clipboard

G = D12.5D4order 192 = 26·3

5th non-split extension by D12 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D12.5D4, Dic6.5D4, M4(2).4D6, C8:D6:7C2, (C2xC4).7D12, (C2xC12).9D4, C4.81(S3xD4), C12.98(C2xD4), (C2xQ8).30D6, D12:C4:4C2, C6.18C22wrC2, C4.10D4:2S3, C3:1(D4.8D4), (C6xQ8).8C22, C12.23D4:1C2, (C2xC12).10C23, Q8.15D6:1C2, C4oD12.6C22, C22.13(C2xD12), C2.21(D6:D4), (C2xD12).41C22, (C4xDic3).2C22, (C3xM4(2)).3C22, (C2xC6).23(C2xD4), (C3xC4.10D4):4C2, (C2xC4).10(C22xS3), SmallGroup(192,312)

Series: Derived Chief Lower central Upper central

C1C2xC12 — D12.5D4
C1C3C6C12C2xC12C4oD12Q8.15D6 — D12.5D4
C3C6C2xC12 — D12.5D4
C1C2C2xC4C4.10D4

Generators and relations for D12.5D4
 G = < a,b,c,d | a12=b2=1, c4=a6, d2=a9, bab=a-1, cac-1=a7, ad=da, cbc-1=a3b, dbd-1=a9b, dcd-1=a9c3 >

Subgroups: 496 in 146 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, C2xC4, D4, Q8, C23, Dic3, C12, C12, D6, C2xC6, C42, C22:C4, M4(2), D8, SD16, C2xD4, C2xQ8, C2xQ8, C4oD4, C24, Dic6, Dic6, C4xS3, D12, D12, C2xDic3, C3:D4, C2xC12, C2xC12, C3xQ8, C22xS3, C4.10D4, C4wrC2, C4.4D4, C8:C22, 2- 1+4, C24:C2, D24, C4xDic3, D6:C4, C3xM4(2), C2xD12, C4oD12, C4oD12, S3xQ8, Q8:3S3, C6xQ8, D4.8D4, D12:C4, C3xC4.10D4, C8:D6, C12.23D4, Q8.15D6, D12.5D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, D12, C22xS3, C22wrC2, C2xD12, S3xD4, D4.8D4, D6:D4, D12.5D4

Character table of D12.5D4

 class 12A2B2C2D2E34A4B4C4D4E4F4G4H6A6B8A8B12A12B12C12D24A24B24C24D
 size 1121212242224412121212248844888888
ρ1111111111111111111111111111    trivial
ρ211111-111111-111-111-1-11111-1-1-1-1    linear of order 2
ρ3111-111111-1-1-11-1-1111-111-1-1-1-111    linear of order 2
ρ4111-11-1111-1-111-1111-1111-1-111-1-1    linear of order 2
ρ5111-1-1-111111-1-1-1-1111111111111    linear of order 2
ρ6111-1-11111111-1-1111-1-11111-1-1-1-1    linear of order 2
ρ71111-1-1111-1-11-111111-111-1-1-1-111    linear of order 2
ρ81111-11111-1-1-1-11-111-1111-1-111-1-1    linear of order 2
ρ9222000-122-2-20000-1-12-2-1-11111-1-1    orthogonal lifted from D6
ρ102220002-2-22-200002200-2-22-20000    orthogonal lifted from D4
ρ11222000-122220000-1-1-2-2-1-1-1-11111    orthogonal lifted from D6
ρ12222000-122-2-20000-1-1-22-1-111-1-111    orthogonal lifted from D6
ρ1322-2-2002-220000202-200-22000000    orthogonal lifted from D4
ρ142220002-2-2-2200002200-2-2-220000    orthogonal lifted from D4
ρ1522-202022-2000-2002-2002-2000000    orthogonal lifted from D4
ρ1622-22002-220000-202-200-22000000    orthogonal lifted from D4
ρ1722-20-2022-20002002-2002-2000000    orthogonal lifted from D4
ρ18222000-122220000-1-122-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ19222000-1-2-22-20000-1-10011-113-3-33    orthogonal lifted from D12
ρ20222000-1-2-22-20000-1-10011-11-333-3    orthogonal lifted from D12
ρ21222000-1-2-2-220000-1-100111-13-33-3    orthogonal lifted from D12
ρ22222000-1-2-2-220000-1-100111-1-33-33    orthogonal lifted from D12
ρ2344-4000-2-44000000-22002-2000000    orthogonal lifted from S3xD4
ρ2444-4000-24-4000000-2200-22000000    orthogonal lifted from S3xD4
ρ254-4000040000-2i002i-400000000000    complex lifted from D4.8D4
ρ264-40000400002i00-2i-400000000000    complex lifted from D4.8D4
ρ278-80000-400000000400000000000    orthogonal faithful, Schur index 2

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

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

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

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

Matrix representation of D12.5D4 in GL6(F73)

0720000
1720000
004606556
00027027
00002760
0000046
,
1720000
0720000
00027027
004606556
00005637
0000817
,
100000
010000
000101
0046000
00001751
00006556
,
100000
010000
0007200
00270817
006025173
001906556

G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,72,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0,0,0,0,65,0,27,0,0,0,56,27,60,46],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,46,0,0,0,0,27,0,0,0,0,0,0,65,56,8,0,0,27,56,37,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,46,0,0,0,0,1,0,0,0,0,0,0,0,17,65,0,0,1,0,51,56],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,27,60,19,0,0,72,0,25,0,0,0,0,8,17,65,0,0,0,17,3,56] >;

D12.5D4 in GAP, Magma, Sage, TeX

D_{12}._5D_4
% in TeX

G:=Group("D12.5D4");
// GroupNames label

G:=SmallGroup(192,312);
// by ID

G=gap.SmallGroup(192,312);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,226,1123,570,136,1684,438,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=1,c^4=a^6,d^2=a^9,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^3*b,d*b*d^-1=a^9*b,d*c*d^-1=a^9*c^3>;
// generators/relations

Export

Character table of D12.5D4 in TeX

׿
x
:
Z
F
o
wr
Q
<