Copied to
clipboard

G = (C3×C12).D4  order 288 = 25·32

2nd non-split extension by C3×C12 of D4 acting faithfully

non-abelian, soluble, monomial

Aliases: C4.10S3≀C2, (C3×C12).2D4, C6.D6.C4, C3⋊Dic3.3D4, C32⋊(C4.10D4), Dic3.D6.1C2, C12.31D6.1C2, C32⋊M4(2).1C2, C2.4(S32⋊C4), (C4×C3⋊S3).2C22, (C3×C6).3(C22⋊C4), (C2×C3⋊S3).7(C2×C4), SmallGroup(288,376)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C2×C3⋊S3 — (C3×C12).D4
Chief series
C1 — C32 — C3×C6 — C2×C3⋊S3 — C4×C3⋊S3 — Dic3.D6 — (C3×C12).D4
Lower central
C32 — C3×C6 — C2×C3⋊S3 — (C3×C12).D4
Upper central
C1 — C2 — C4

Generators and relations for (C3×C12).D4
 G = < a,b,c,d | a3=b12=1, c4=d2=b6, ab=ba, cac-1=b4, dad-1=a-1, cbc-1=a-1b3, dbd-1=b7, dcd-1=b9c3 >

Subgroups: 328 in 67 conjugacy classes, 15 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C2×C4, Q8, C32, Dic3, C12, D6, M4(2), C2×Q8, C3⋊S3, C3×C6, C3⋊C8, C24, Dic6, C4×S3, C3×Q8, C4.10D4, C3×Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C8⋊S3, S3×Q8, C3×C3⋊C8, C32⋊2C8, C6.D6, C32⋊2Q8, C3×Dic6, C4×C3⋊S3, C12.31D6, C32⋊M4(2), Dic3.D6, (C3×C12).D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, C4.10D4, S3≀C2, S32⋊C4, (C3×C12).D4

Character table of (C3×C12).D4

 class 12A2B3A3B4A4B4C4D6A6B8A8B8C8D12A12B12C12D12E24A24B24C24D
 size 11184421212184412123636448242412121212
ρ1111111111111111111111111    trivial
ρ2111111-1-1111-1-111111-1-1-1-1-1-1    linear of order 2
ρ311111111111-1-1-1-111111-1-1-1-1    linear of order 2
ρ4111111-1-111111-1-1111-1-11111    linear of order 2
ρ511111-11-1-111-ii-ii-1-1-1-11i-i-ii    linear of order 4
ρ611111-1-11-111i-i-ii-1-1-11-1-iii-i    linear of order 4
ρ711111-11-1-111i-ii-i-1-1-1-11-iii-i    linear of order 4
ρ811111-1-11-111-iii-i-1-1-11-1i-i-ii    linear of order 4
ρ922-222200-2220000222000000    orthogonal lifted from D4
ρ1022-222-2002220000-2-2-2000000    orthogonal lifted from D4
ρ11440-214000-21220011-200-1-1-1-1    orthogonal lifted from S3≀C2
ρ124401-24-2-201-20000-2-21110000    orthogonal lifted from S3≀C2
ρ134401-2-42-201-2000022-11-10000    orthogonal lifted from S32⋊C4
ρ14440-214000-21-2-20011-2001111    orthogonal lifted from S3≀C2
ρ154401-2-4-2201-2000022-1-110000    orthogonal lifted from S32⋊C4
ρ164401-242201-20000-2-21-1-10000    orthogonal lifted from S3≀C2
ρ174-40440000-4-40000000000000    symplectic lifted from C4.10D4, Schur index 2
ρ18440-21-4000-212i-2i00-1-1200i-i-ii    complex lifted from S32⋊C4
ρ19440-21-4000-21-2i2i00-1-1200-iii-i    complex lifted from S32⋊C4
ρ204-40-2100002-100003i-3i0002ζ8ζ3+ζ82ζ87ζ3+ζ872ζ83ζ3+ζ832ζ85ζ3+ζ85    complex faithful
ρ214-40-2100002-10000-3i3i0002ζ87ζ3+ζ872ζ8ζ3+ζ82ζ85ζ3+ζ852ζ83ζ3+ζ83    complex faithful
ρ224-40-2100002-100003i-3i0002ζ85ζ3+ζ852ζ83ζ3+ζ832ζ87ζ3+ζ872ζ8ζ3+ζ8    complex faithful
ρ234-40-2100002-10000-3i3i0002ζ83ζ3+ζ832ζ85ζ3+ζ852ζ8ζ3+ζ82ζ87ζ3+ζ87    complex faithful
ρ248-802-40000-240000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of (C3×C12).D4
►On 48 points
Generators in S48
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)

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

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

Matrix representation of (C3×C12).D4 ►in GL4(𝔽5) generated by

1100
2300
0011
0023
,
3300
1400
0013
0012
,
0033
0002
1000
0100
,
0043
0021
3100
4200
G:=sub<GL(4,GF(5))| [1,2,0,0,1,3,0,0,0,0,1,2,0,0,1,3],[3,1,0,0,3,4,0,0,0,0,1,1,0,0,3,2],[0,0,1,0,0,0,0,1,3,0,0,0,3,2,0,0],[0,0,3,4,0,0,1,2,4,2,0,0,3,1,0,0] >;
 

(C3×C12).D4 in GAP, Magma, Sage, TeX 

(C_3\times C_{12}).D_4
 
% in TeX
 
G:=Group("(C3xC12).D4");
 
// GroupNames label
 
G:=SmallGroup(288,376);
 
// by ID
 
G=gap.SmallGroup(288,376);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,112,85,64,219,675,80,2693,2028,691,797,2372]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^12=1,c^4=d^2=b^6,a*b=b*a,c*a*c^-1=b^4,d*a*d^-1=a^-1,c*b*c^-1=a^-1*b^3,d*b*d^-1=b^7,d*c*d^-1=b^9*c^3>;
 
// generators/relations
 

Export

Character table of (C3×C12).D4 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁