Copied to
clipboard

G = (C3×C6).9D12  order 432 = 24·33

2nd non-split extension by C3×C6 of D12 acting via D12/C3=D4

non-abelian, soluble, monomial

Aliases: C6.8S3≀C2, C33⋊4(C4⋊C4), (C3×C6).9D12, C3⋊S3.2Dic6, C32⋊C4⋊1Dic3, (C32×C6).14D4, C32⋊2(C4⋊Dic3), C2.2(C32⋊2D12), (C3×C32⋊C4)⋊1C4, (C3×C3⋊S3).4Q8, (C2×C3⋊S3).14D6, C3⋊2(C3⋊S3.Q8), (C2×C32⋊C4).2S3, (C6×C32⋊C4).5C2, C33⋊9(C2×C4).4C2, C3⋊S3.3(C2×Dic3), (C6×C3⋊S3).10C22, (C3×C3⋊S3).12(C2×C4), SmallGroup(432,587)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3×C3⋊S3 — (C3×C6).9D12
Chief series
C1 — C3 — C33 — C3×C3⋊S3 — C6×C3⋊S3 — C33⋊9(C2×C4) — (C3×C6).9D12
Lower central
C33 — C3×C3⋊S3 — (C3×C6).9D12
Upper central
C1 — C2

Generators and relations for (C3×C6).9D12
 G = < a,b,c,d | a3=b6=c12=1, d2=b3, ab=ba, cac-1=b2, dad-1=a-1, cbc-1=ab3, bd=db, dcd-1=c-1 >

Subgroups: 592 in 96 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, C32, C32, Dic3, C12, D6, C2×C6, C4⋊C4, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C32⋊C4, S3×C6, C2×C3⋊S3, C4⋊Dic3, C3×C3⋊S3, C32×C6, S3×Dic3, C6.D6, C2×C32⋊C4, C3×C3⋊Dic3, C3×C32⋊C4, C6×C3⋊S3, C3⋊S3.Q8, C33⋊9(C2×C4), C6×C32⋊C4, (C3×C6).9D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, Dic6, D12, C2×Dic3, C4⋊Dic3, S3≀C2, C3⋊S3.Q8, C32⋊2D12, (C3×C6).9D12

Character table of (C3×C6).9D12

 class 12A2B2C3A3B3C3D3E4A4B4C4D4E4F6A6B6C6D6E6F6G12A12B12C12D12E12F12G12H
 size 1199244881818181818182448818181818181836363636
ρ1111111111111111111111111111111    trivial
ρ2111111111-111-1-1-111111111111-1-1-1-1    linear of order 2
ρ3111111111-1-1-111-11111111-1-1-1-11-11-1    linear of order 2
ρ41111111111-1-1-1-111111111-1-1-1-1-11-11    linear of order 2
ρ51-1-1111111-i-11-iii-1-1-1-1-1-11-1-111-iii-i    linear of order 4
ρ61-1-1111111i-11i-i-i-1-1-1-1-1-11-1-111i-i-ii    linear of order 4
ρ71-1-1111111i1-1-ii-i-1-1-1-1-1-1111-1-1-i-iii    linear of order 4
ρ81-1-1111111-i1-1i-ii-1-1-1-1-1-1111-1-1ii-i-i    linear of order 4
ρ92222-122-1-10-2-2000-122-1-1-1-111110000    orthogonal lifted from D6
ρ1022-2-22222200000022222-2-200000000    orthogonal lifted from D4
ρ112222-122-1-1022000-122-1-1-1-1-1-1-1-10000    orthogonal lifted from S3
ρ1222-2-2-122-1-1000000-122-1-111-√3√3√3-√30000    orthogonal lifted from D12
ρ1322-2-2-122-1-1000000-122-1-111√3-√3-√3√30000    orthogonal lifted from D12
ρ142-2-22-122-1-10-220001-2-2111-111-1-10000    symplectic lifted from Dic3, Schur index 2
ρ152-2-22-122-1-102-20001-2-2111-1-1-1110000    symplectic lifted from Dic3, Schur index 2
ρ162-22-222222000000-2-2-2-2-22-200000000    symplectic lifted from Q8, Schur index 2
ρ172-22-2-122-1-10000001-2-211-11√3-√3√3-√30000    symplectic lifted from Dic6, Schur index 2
ρ182-22-2-122-1-10000001-2-211-11-√3√3-√3√30000    symplectic lifted from Dic6, Schur index 2
ρ19440041-21-200022041-21-2000000-10-10    orthogonal lifted from S3≀C2
ρ2044004-21-21-20000-24-21-210000000101    orthogonal lifted from S3≀C2
ρ21440041-21-2000-2-2041-21-20000001010    orthogonal lifted from S3≀C2
ρ2244004-21-212000024-21-210000000-10-1    orthogonal lifted from S3≀C2
ρ234-40041-21-20002i-2i0-4-12-12000000-i0i0    complex lifted from C3⋊S3.Q8
ρ244-40041-21-2000-2i2i0-4-12-12000000i0-i0    complex lifted from C3⋊S3.Q8
ρ254-4004-21-21-2i00002i-42-12-10000000-i0i    complex lifted from C3⋊S3.Q8
ρ264-4004-21-212i0000-2i-42-12-10000000i0-i    complex lifted from C3⋊S3.Q8
ρ278800-4-422-1000000-4-422-10000000000    orthogonal lifted from C32⋊2D12
ρ288800-42-4-12000000-42-4-120000000000    orthogonal lifted from C32⋊2D12
ρ298-800-4-422-100000044-2-210000000000    symplectic faithful, Schur index 2
ρ308-800-42-4-120000004-241-20000000000    symplectic faithful, Schur index 2

Smallest permutation representation of (C3×C6).9D12
►On 48 points
Generators in S48
(1 5 9)(3 11 7)(14 22 18)(16 20 24)(25 33 29)(27 31 35)(37 45 41)(39 43 47)

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

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

Matrix representation of (C3×C6).9D12 ►in GL6(𝔽13)

100000
010000
001000
0000012
000010
0001012
,
1200000
0120000
0010120
0001200
001000
0000012
,
10100000
370000
0001012
001000
0000012
000010
,
220000
4110000
005000
000508
000050
000008

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,12,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[10,3,0,0,0,0,10,7,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,12,0,12,0],[2,4,0,0,0,0,2,11,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,8,0,8] >;
 

(C3×C6).9D12 in GAP, Magma, Sage, TeX 

(C_3\times C_6)._9D_{12}
 
% in TeX
 
G:=Group("(C3xC6).9D12");
 
// GroupNames label
 
G:=SmallGroup(432,587);
 
// by ID
 
G=gap.SmallGroup(432,587);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,85,36,1684,1691,298,677,348,1027,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^6=c^12=1,d^2=b^3,a*b=b*a,c*a*c^-1=b^2,d*a*d^-1=a^-1,c*b*c^-1=a*b^3,b*d=d*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

Export

Character table of (C3×C6).9D12 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁