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G = D12.22D6  order 288 = 25·32

7th non-split extension by D12 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial

Aliases: D12.22D6, Dic6.6D6, D4.4S32, D4:S3:7S3, C3:C8.12D6, D4:2S3:2S3, (C3xD4).21D6, (S3xC6).10D4, (C4xS3).20D6, C6.151(S3xD4), D12:5S3:5C2, C3:5(D8:3S3), D6.1(C3:D4), C32:10(C4oD8), C32:3Q16:7C2, Dic6:S3:9C2, C3:2(Q8.13D6), C32:9SD16:4C2, (C3xC12).10C23, C12.10(C22xS3), (C3xDic3).39D4, (S3xC12).15C22, (C3xD12).13C22, C32:4C8.7C22, (D4xC32).6C22, Dic3.20(C3:D4), C32:4Q8.8C22, (C3xDic6).12C22, (S3xC3:C8):2C2, C4.10(C2xS32), (C3xD4:S3):3C2, C6.47(C2xC3:D4), C2.25(S3xC3:D4), (C3xC3:C8).5C22, (C3xD4:2S3):2C2, (C3xC6).125(C2xD4), SmallGroup(288,581)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3xC12 — D12.22D6
Chief series
C1 — C3 — C32 — C3xC6 — C3xC12 — S3xC12 — D12:5S3 — D12.22D6
Lower central
C32 — C3xC6 — C3xC12 — D12.22D6
Upper central
C1 — C2 — C4 — D4

Generators and relations for D12.22D6
 G = < a,b,c,d | a12=b2=c6=1, d2=a6, bab=a-1, cac-1=dad-1=a7, cbc-1=a3b, dbd-1=a9b, dcd-1=a6c-1 >

Subgroups: 490 in 134 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2xC4, D4, D4, Q8, C32, Dic3, Dic3, C12, C12, D6, D6, C2xC6, C2xC8, D8, SD16, Q16, C4oD4, C3xS3, C3xC6, C3xC6, C3:C8, C3:C8, C24, Dic6, Dic6, C4xS3, C4xS3, D12, C2xDic3, C3:D4, C2xC12, C3xD4, C3xD4, C3xQ8, C4oD8, C3xDic3, C3xDic3, C3:Dic3, C3xC12, S3xC6, S3xC6, C62, S3xC8, Dic12, C2xC3:C8, D4:S3, D4.S3, Q8:2S3, C3:Q16, C3xD8, C4oD12, D4:2S3, D4:2S3, C3xC4oD4, C3xC3:C8, C32:4C8, S3xDic3, D6:S3, C3xDic6, S3xC12, C3xD12, C6xDic3, C3xC3:D4, C32:4Q8, D4xC32, D8:3S3, Q8.13D6, S3xC3:C8, Dic6:S3, C32:3Q16, C3xD4:S3, C32:9SD16, D12:5S3, C3xD4:2S3, D12.22D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C3:D4, C22xS3, C4oD8, S32, S3xD4, C2xC3:D4, C2xS32, D8:3S3, Q8.13D6, S3xC3:D4, D12.22D6

Smallest permutation representation of D12.22D6
►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 15)(2 14)(3 13)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)(25 42)(26 41)(27 40)(28 39)(29 38)(30 37)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)

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

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

36 conjugacy classes

class 1 2A2B2C2D3A3B3C4A4B4C4D4E6A6B6C6D6E6F6G6H6I6J8A8B8C8D12A12B12C12D12E12F12G24A24B
order122223334444466666666668888121212121212122424
size11461222423312362244488812246618184466812121212

36 irreducible representations

dim111111112222222222224444448
type++++++++++++++++++++--
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6D6D6C3:D4C3:D4C4oD8S32S3xD4C2xS32D8:3S3Q8.13D6S3xC3:D4D12.22D6
kernelD12.22D6S3xC3:C8Dic6:S3C32:3Q16C3xD4:S3C32:9SD16D12:5S3C3xD4:2S3D4:S3D4:2S3C3xDic3S3xC6C3:C8Dic6C4xS3D12C3xD4Dic3D6C32D4C6C4C3C3C2C1
# reps111111111111111122241112221

Matrix representation of D12.22D6 ►in GL6(F73)

010000
7200000
001000
000100
0000721
0000720
,
57570000
57160000
001000
000100
0000720
0000721
,
100000
0720000
0017200
001000
000010
000001
,
0460000
4600000
001000
0017200
000010
000001

G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[57,57,0,0,0,0,57,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,1],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,46,0,0,0,0,46,0,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

D12.22D6 in GAP, Magma, Sage, TeX 

D_{12}._{22}D_6
 
% in TeX
 
G:=Group("D12.22D6");
 
// GroupNames label
 
G:=SmallGroup(288,581);
 
// by ID
 
G=gap.SmallGroup(288,581);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,422,135,346,185,80,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^12=b^2=c^6=1,d^2=a^6,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^7,c*b*c^-1=a^3*b,d*b*d^-1=a^9*b,d*c*d^-1=a^6*c^-1>;
 
// generators/relations
 

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