Copied to
clipboard

G = C12.47D4order 96 = 25·3

4th non-split extension by C12 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.47D4, C4.12D12, M4(2).2S3, (C2×C4).2D6, (C2×Dic3).C4, C22.5(C4×S3), C2.10(D6⋊C4), C4.22(C3⋊D4), C31(C4.10D4), C6.9(C22⋊C4), (C2×Dic6).6C2, C4.Dic3.3C2, (C2×C12).14C22, (C3×M4(2)).2C2, (C2×C6).3(C2×C4), SmallGroup(96,31)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C12.47D4
C1C3C6C12C2×C12C2×Dic6 — C12.47D4
C3C6C2×C6 — C12.47D4
C1C2C2×C4M4(2)

Generators and relations for C12.47D4
 G = < a,b,c | a12=1, b4=c2=a6, bab-1=cac-1=a-1, cbc-1=a9b3 >

2C2
6C4
6C4
2C6
2C8
3C2×C4
3C2×C4
6C8
6Q8
6Q8
2Dic3
2Dic3
3M4(2)
3C2×Q8
2Dic6
2Dic6
2C3⋊C8
2C24
3C4.10D4

Character table of C12.47D4

 class 12A2B34A4B4C4D6A6B8A8B8C8D12A12B12C24A24B24C24D
 size 1122221212244412122244444
ρ1111111111111111111111    trivial
ρ2111111-1-111-1-111111-1-1-1-1    linear of order 2
ρ31111111111-1-1-1-1111-1-1-1-1    linear of order 2
ρ4111111-1-11111-1-11111111    linear of order 2
ρ51111-1-11-111-iii-i-1-1-1-i-iii    linear of order 4
ρ61111-1-1-1111i-ii-i-1-1-1ii-i-i    linear of order 4
ρ71111-1-11-111i-i-ii-1-1-1ii-i-i    linear of order 4
ρ81111-1-1-1111-ii-ii-1-1-1-i-iii    linear of order 4
ρ922-22-22002-20000-2-220000    orthogonal lifted from D4
ρ10222-12200-1-1-2-200-1-1-11111    orthogonal lifted from D6
ρ11222-12200-1-12200-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1222-222-2002-2000022-20000    orthogonal lifted from D4
ρ1322-2-1-2200-11000011-13-3-33    orthogonal lifted from D12
ρ1422-2-1-2200-11000011-1-333-3    orthogonal lifted from D12
ρ15222-1-2-200-1-12i-2i00111-i-iii    complex lifted from C4×S3
ρ16222-1-2-200-1-1-2i2i00111ii-i-i    complex lifted from C4×S3
ρ1722-2-12-200-110000-1-11--3-3--3-3    complex lifted from C3⋊D4
ρ1822-2-12-200-110000-1-11-3--3-3--3    complex lifted from C3⋊D4
ρ194-4040000-4000000000000    symplectic lifted from C4.10D4, Schur index 2
ρ204-40-20000200000-232300000    symplectic faithful, Schur index 2
ρ214-40-2000020000023-2300000    symplectic faithful, Schur index 2

Smallest permutation representation of C12.47D4
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 43 10 46 7 37 4 40)(2 42 11 45 8 48 5 39)(3 41 12 44 9 47 6 38)(13 26 16 35 19 32 22 29)(14 25 17 34 20 31 23 28)(15 36 18 33 21 30 24 27)
(1 25 7 31)(2 36 8 30)(3 35 9 29)(4 34 10 28)(5 33 11 27)(6 32 12 26)(13 47 19 41)(14 46 20 40)(15 45 21 39)(16 44 22 38)(17 43 23 37)(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), (1,43,10,46,7,37,4,40)(2,42,11,45,8,48,5,39)(3,41,12,44,9,47,6,38)(13,26,16,35,19,32,22,29)(14,25,17,34,20,31,23,28)(15,36,18,33,21,30,24,27), (1,25,7,31)(2,36,8,30)(3,35,9,29)(4,34,10,28)(5,33,11,27)(6,32,12,26)(13,47,19,41)(14,46,20,40)(15,45,21,39)(16,44,22,38)(17,43,23,37)(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), (1,43,10,46,7,37,4,40)(2,42,11,45,8,48,5,39)(3,41,12,44,9,47,6,38)(13,26,16,35,19,32,22,29)(14,25,17,34,20,31,23,28)(15,36,18,33,21,30,24,27), (1,25,7,31)(2,36,8,30)(3,35,9,29)(4,34,10,28)(5,33,11,27)(6,32,12,26)(13,47,19,41)(14,46,20,40)(15,45,21,39)(16,44,22,38)(17,43,23,37)(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)], [(1,43,10,46,7,37,4,40),(2,42,11,45,8,48,5,39),(3,41,12,44,9,47,6,38),(13,26,16,35,19,32,22,29),(14,25,17,34,20,31,23,28),(15,36,18,33,21,30,24,27)], [(1,25,7,31),(2,36,8,30),(3,35,9,29),(4,34,10,28),(5,33,11,27),(6,32,12,26),(13,47,19,41),(14,46,20,40),(15,45,21,39),(16,44,22,38),(17,43,23,37),(18,42,24,48)])

C12.47D4 is a maximal subgroup of
M4(2).19D6  D12.2D4  S3×C4.10D4  D12.7D4  Q8.14D12  D4.10D12  C24.18D4  C24.42D4  M4(2).31D6  Q8.8D12  Q8.10D12  M4(2).13D6  D12.38D4  M4(2).16D6  D12.40D4  C4.D36  C12.14D12  C12.71D12  C12.20D12  C60.54D4  C60.31D4  C4.D60  Dic5.4D12
C12.47D4 is a maximal quotient of
C42.2D6  (C2×Dic3)⋊C8  C12.47D8  C12.2D8  M4(2)⋊Dic3  C4.D36  C12.14D12  C12.71D12  C12.20D12  C60.54D4  C60.31D4  C4.D60  Dic5.4D12

Matrix representation of C12.47D4 in GL4(𝔽73) generated by

666600
75900
006666
00759
,
007120
00182
473400
82600
,
712000
18200
007120
00182
G:=sub<GL(4,GF(73))| [66,7,0,0,66,59,0,0,0,0,66,7,0,0,66,59],[0,0,47,8,0,0,34,26,71,18,0,0,20,2,0,0],[71,18,0,0,20,2,0,0,0,0,71,18,0,0,20,2] >;

C12.47D4 in GAP, Magma, Sage, TeX

C_{12}._{47}D_4
% in TeX

G:=Group("C12.47D4");
// GroupNames label

G:=SmallGroup(96,31);
// by ID

G=gap.SmallGroup(96,31);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,121,31,362,86,297,2309]);
// Polycyclic

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

Export

Subgroup lattice of C12.47D4 in TeX
Character table of C12.47D4 in TeX

׿
×
𝔽