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G = C12.82D12  order 288 = 25·32

13rd non-split extension by C12 of D12 acting via D12/D6=C2

metabelian, supersoluble, monomial

Aliases: C12.82D12, C62.4Q8, C3:C8.1Dic3, C12.91(C4xS3), (C2xC6).7Dic6, (C3xC12).114D4, (C2xC12).288D6, C6.4(C4:Dic3), C4.13(S3xDic3), C3:1(C24.C4), C4.Dic3.1S3, C12.81(C3:D4), C6.7(Dic3:C4), (C6xC12).37C22, C32:5(C8.C4), C12.12(C2xDic3), C4.31(C3:D12), C3:1(C12.53D4), C12.58D6.3C2, C2.5(Dic3:Dic3), C22.1(C32:2Q8), (C3xC3:C8).1C4, (C2xC4).60S32, (C2xC3:C8).5S3, (C6xC3:C8).11C2, (C3xC6).24(C4:C4), (C3xC12).35(C2xC4), (C3xC4.Dic3).3C2, SmallGroup(288,225)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3xC12 — C12.82D12
Chief series
C1 — C3 — C32 — C3xC6 — C3xC12 — C6xC12 — C6xC3:C8 — C12.82D12
Lower central
C32 — C3xC6 — C3xC12 — C12.82D12
Upper central
C1 — C4 — C2xC4

Generators and relations for C12.82D12
 G = < a,b,c | a12=1, b12=a6, c2=a9, bab-1=cac-1=a5, cbc-1=a6b11 >

Subgroups: 178 in 70 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C8, C2xC4, C32, C12, C12, C2xC6, C2xC6, C2xC8, M4(2), C3xC6, C3xC6, C3:C8, C3:C8, C24, C2xC12, C2xC12, C8.C4, C3xC12, C62, C2xC3:C8, C4.Dic3, C4.Dic3, C2xC24, C3xM4(2), C3xC3:C8, C3xC3:C8, C32:4C8, C6xC12, C24.C4, C12.53D4, C6xC3:C8, C3xC4.Dic3, C12.58D6, C12.82D12
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, Q8, Dic3, D6, C4:C4, Dic6, C4xS3, D12, C2xDic3, C3:D4, C8.C4, S32, Dic3:C4, C4:Dic3, S3xDic3, C3:D12, C32:2Q8, C24.C4, C12.53D4, Dic3:Dic3, C12.82D12

Smallest permutation representation of C12.82D12
►On 48 points
Generators in S48
(1 23 21 19 17 15 13 11 9 7 5 3)(2 16 6 20 10 24 14 4 18 8 22 12)(25 35 45 31 41 27 37 47 33 43 29 39)(26 28 30 32 34 36 38 40 42 44 46 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 39 7 33 13 27 19 45)(2 38 8 32 14 26 20 44)(3 37 9 31 15 25 21 43)(4 36 10 30 16 48 22 42)(5 35 11 29 17 47 23 41)(6 34 12 28 18 46 24 40)
 
G:=sub<Sym(48)| (1,23,21,19,17,15,13,11,9,7,5,3)(2,16,6,20,10,24,14,4,18,8,22,12)(25,35,45,31,41,27,37,47,33,43,29,39)(26,28,30,32,34,36,38,40,42,44,46,48), (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), (1,39,7,33,13,27,19,45)(2,38,8,32,14,26,20,44)(3,37,9,31,15,25,21,43)(4,36,10,30,16,48,22,42)(5,35,11,29,17,47,23,41)(6,34,12,28,18,46,24,40)>;
 
G:=Group( (1,23,21,19,17,15,13,11,9,7,5,3)(2,16,6,20,10,24,14,4,18,8,22,12)(25,35,45,31,41,27,37,47,33,43,29,39)(26,28,30,32,34,36,38,40,42,44,46,48), (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), (1,39,7,33,13,27,19,45)(2,38,8,32,14,26,20,44)(3,37,9,31,15,25,21,43)(4,36,10,30,16,48,22,42)(5,35,11,29,17,47,23,41)(6,34,12,28,18,46,24,40) );
 
G=PermutationGroup([[(1,23,21,19,17,15,13,11,9,7,5,3),(2,16,6,20,10,24,14,4,18,8,22,12),(25,35,45,31,41,27,37,47,33,43,29,39),(26,28,30,32,34,36,38,40,42,44,46,48)], [(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)], [(1,39,7,33,13,27,19,45),(2,38,8,32,14,26,20,44),(3,37,9,31,15,25,21,43),(4,36,10,30,16,48,22,42),(5,35,11,29,17,47,23,41),(6,34,12,28,18,46,24,40)]])
 

48 conjugacy classes

class 1 2A2B3A3B3C4A4B4C6A6B6C6D6E6F6G6H8A8B8C8D8E8F8G8H12A···12F12G···12K24A···24H24I24J24K24L
order122333444666666668888888812···1212···1224···2424242424
size112224112222244446666121236362···24···46···612121212

48 irreducible representations

dim11111222222222222444444
type+++++++--++-+-+-
imageC1C2C2C2C4S3S3D4Q8Dic3D6C4xS3D12C3:D4Dic6C8.C4C24.C4S32S3xDic3C3:D12C32:2Q8C12.53D4C12.82D12
kernelC12.82D12C6xC3:C8C3xC4.Dic3C12.58D6C3xC3:C8C2xC3:C8C4.Dic3C3xC12C62C3:C8C2xC12C12C12C12C2xC6C32C3C2xC4C4C4C22C3C1
# reps11114111122222448111124

Matrix representation of C12.82D12 ►in GL6(F73)

4600000
0460000
000100
0072100
0000720
0000072
,
1000000
0220000
000100
001000
0000721
0000720
,
0220000
2200000
0004600
0046000
0000270
00002746

G:=sub<GL(6,GF(73))| [46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[10,0,0,0,0,0,0,22,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[0,22,0,0,0,0,22,0,0,0,0,0,0,0,0,46,0,0,0,0,46,0,0,0,0,0,0,0,27,27,0,0,0,0,0,46] >;
 

C12.82D12 in GAP, Magma, Sage, TeX 

C_{12}._{82}D_{12}
 
% in TeX
 
G:=Group("C12.82D12");
 
// GroupNames label
 
G:=SmallGroup(288,225);
 
// by ID
 
G=gap.SmallGroup(288,225);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,141,36,100,346,80,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c|a^12=1,b^12=a^6,c^2=a^9,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=a^6*b^11>;
 
// generators/relations
 

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