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G = (C3×C6).9D12order 432 = 24·33

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

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 [×2], C3, C3 [×4], C4 [×4], C22, S3 [×4], C6, C6 [×6], C2×C4 [×3], C32, C32 [×4], Dic3 [×6], C12 [×4], D6 [×2], C2×C6, C4⋊C4, C3×S3 [×4], C3⋊S3 [×2], C3×C6, C3×C6 [×4], C4×S3 [×2], C2×Dic3 [×2], C2×C12, C33, C3×Dic3 [×6], C3⋊Dic3 [×2], C32⋊C4 [×2], S3×C6 [×2], C2×C3⋊S3, C4⋊Dic3, C3×C3⋊S3 [×2], C32×C6, S3×Dic3 [×2], C6.D6 [×2], C2×C32⋊C4, C3×C3⋊Dic3 [×2], C3×C32⋊C4 [×2], C6×C3⋊S3, C3⋊S3.Q8, C339(C2×C4) [×2], C6×C32⋊C4, (C3×C6).9D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4, Q8, Dic3 [×2], D6, C4⋊C4, Dic6, D12, C2×Dic3, C4⋊Dic3, S3≀C2, C3⋊S3.Q8, C322D12, (C3×C6).9D12

Character table of (C3×C6).9D12

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 9 9 2 4 4 8 8 18 18 18 18 18 18 2 4 4 8 8 18 18 18 18 18 18 36 36 36 36 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 -1 1 1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 -1 -1 -1 1 1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 linear of order 2 ρ5 1 -1 -1 1 1 1 1 1 1 -i -1 1 -i i i -1 -1 -1 -1 -1 -1 1 -1 -1 1 1 -i i i -i linear of order 4 ρ6 1 -1 -1 1 1 1 1 1 1 i -1 1 i -i -i -1 -1 -1 -1 -1 -1 1 -1 -1 1 1 i -i -i i linear of order 4 ρ7 1 -1 -1 1 1 1 1 1 1 i 1 -1 -i i -i -1 -1 -1 -1 -1 -1 1 1 1 -1 -1 -i -i i i linear of order 4 ρ8 1 -1 -1 1 1 1 1 1 1 -i 1 -1 i -i i -1 -1 -1 -1 -1 -1 1 1 1 -1 -1 i i -i -i linear of order 4 ρ9 2 2 2 2 -1 2 2 -1 -1 0 -2 -2 0 0 0 -1 2 2 -1 -1 -1 -1 1 1 1 1 0 0 0 0 orthogonal lifted from D6 ρ10 2 2 -2 -2 2 2 2 2 2 0 0 0 0 0 0 2 2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -1 2 2 -1 -1 0 2 2 0 0 0 -1 2 2 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ12 2 2 -2 -2 -1 2 2 -1 -1 0 0 0 0 0 0 -1 2 2 -1 -1 1 1 -√3 √3 √3 -√3 0 0 0 0 orthogonal lifted from D12 ρ13 2 2 -2 -2 -1 2 2 -1 -1 0 0 0 0 0 0 -1 2 2 -1 -1 1 1 √3 -√3 -√3 √3 0 0 0 0 orthogonal lifted from D12 ρ14 2 -2 -2 2 -1 2 2 -1 -1 0 -2 2 0 0 0 1 -2 -2 1 1 1 -1 1 1 -1 -1 0 0 0 0 symplectic lifted from Dic3, Schur index 2 ρ15 2 -2 -2 2 -1 2 2 -1 -1 0 2 -2 0 0 0 1 -2 -2 1 1 1 -1 -1 -1 1 1 0 0 0 0 symplectic lifted from Dic3, Schur index 2 ρ16 2 -2 2 -2 2 2 2 2 2 0 0 0 0 0 0 -2 -2 -2 -2 -2 2 -2 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ17 2 -2 2 -2 -1 2 2 -1 -1 0 0 0 0 0 0 1 -2 -2 1 1 -1 1 √3 -√3 √3 -√3 0 0 0 0 symplectic lifted from Dic6, Schur index 2 ρ18 2 -2 2 -2 -1 2 2 -1 -1 0 0 0 0 0 0 1 -2 -2 1 1 -1 1 -√3 √3 -√3 √3 0 0 0 0 symplectic lifted from Dic6, Schur index 2 ρ19 4 4 0 0 4 1 -2 1 -2 0 0 0 2 2 0 4 1 -2 1 -2 0 0 0 0 0 0 -1 0 -1 0 orthogonal lifted from S3≀C2 ρ20 4 4 0 0 4 -2 1 -2 1 -2 0 0 0 0 -2 4 -2 1 -2 1 0 0 0 0 0 0 0 1 0 1 orthogonal lifted from S3≀C2 ρ21 4 4 0 0 4 1 -2 1 -2 0 0 0 -2 -2 0 4 1 -2 1 -2 0 0 0 0 0 0 1 0 1 0 orthogonal lifted from S3≀C2 ρ22 4 4 0 0 4 -2 1 -2 1 2 0 0 0 0 2 4 -2 1 -2 1 0 0 0 0 0 0 0 -1 0 -1 orthogonal lifted from S3≀C2 ρ23 4 -4 0 0 4 1 -2 1 -2 0 0 0 2i -2i 0 -4 -1 2 -1 2 0 0 0 0 0 0 -i 0 i 0 complex lifted from C3⋊S3.Q8 ρ24 4 -4 0 0 4 1 -2 1 -2 0 0 0 -2i 2i 0 -4 -1 2 -1 2 0 0 0 0 0 0 i 0 -i 0 complex lifted from C3⋊S3.Q8 ρ25 4 -4 0 0 4 -2 1 -2 1 -2i 0 0 0 0 2i -4 2 -1 2 -1 0 0 0 0 0 0 0 -i 0 i complex lifted from C3⋊S3.Q8 ρ26 4 -4 0 0 4 -2 1 -2 1 2i 0 0 0 0 -2i -4 2 -1 2 -1 0 0 0 0 0 0 0 i 0 -i complex lifted from C3⋊S3.Q8 ρ27 8 8 0 0 -4 -4 2 2 -1 0 0 0 0 0 0 -4 -4 2 2 -1 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊2D12 ρ28 8 8 0 0 -4 2 -4 -1 2 0 0 0 0 0 0 -4 2 -4 -1 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊2D12 ρ29 8 -8 0 0 -4 -4 2 2 -1 0 0 0 0 0 0 4 4 -2 -2 1 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ30 8 -8 0 0 -4 2 -4 -1 2 0 0 0 0 0 0 4 -2 4 1 -2 0 0 0 0 0 0 0 0 0 0 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 29 33)(27 35 31)(37 45 41)(39 43 47)
(1 24)(2 17 10 13 6 21)(3 14)(4 23 8 15 12 19)(5 16)(7 18)(9 20)(11 22)(25 39)(26 44 34 40 30 48)(27 41)(28 38 32 42 36 46)(29 43)(31 45)(33 47)(35 37)
(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 24 29)(2 42 13 28)(3 41 14 27)(4 40 15 26)(5 39 16 25)(6 38 17 36)(7 37 18 35)(8 48 19 34)(9 47 20 33)(10 46 21 32)(11 45 22 31)(12 44 23 30)

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

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

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

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

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1 0 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 12 0 0 0 0 12 0 0 0 0 1 0 0 0 0 0 0 0 0 12
,
 10 10 0 0 0 0 3 7 0 0 0 0 0 0 0 1 0 12 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 1 0
,
 2 2 0 0 0 0 4 11 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 8 0 0 0 0 5 0 0 0 0 0 0 8

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

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