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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, (C3×D4).21D6, (S3×C6).10D4, (C4×S3).20D6, C6.151(S3×D4), D12⋊5S3⋊5C2, C3⋊5(D8⋊3S3), D6.1(C3⋊D4), C32⋊10(C4○D8), C32⋊3Q16⋊7C2, Dic6⋊S3⋊9C2, C3⋊2(Q8.13D6), C32⋊9SD16⋊4C2, (C3×C12).10C23, C12.10(C22×S3), (C3×Dic3).39D4, (S3×C12).15C22, (C3×D12).13C22, C32⋊4C8.7C22, (D4×C32).6C22, Dic3.20(C3⋊D4), C32⋊4Q8.8C22, (C3×Dic6).12C22, (S3×C3⋊C8)⋊2C2, C4.10(C2×S32), (C3×D4⋊S3)⋊3C2, C6.47(C2×C3⋊D4), C2.25(S3×C3⋊D4), (C3×C3⋊C8).5C22, (C3×D4⋊2S3)⋊2C2, (C3×C6).125(C2×D4), SmallGroup(288,581)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — D12.22D6
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — D12⋊5S3 — D12.22D6
Lower central
C32 — C3×C6 — C3×C12 — 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, C2×C4, D4, D4, Q8, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, C4○D8, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C62, S3×C8, Dic12, C2×C3⋊C8, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C3×D8, C4○D12, D4⋊2S3, D4⋊2S3, C3×C4○D4, C3×C3⋊C8, C32⋊4C8, S3×Dic3, D6⋊S3, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C32⋊4Q8, D4×C32, D8⋊3S3, Q8.13D6, S3×C3⋊C8, Dic6⋊S3, C32⋊3Q16, C3×D4⋊S3, C32⋊9SD16, D12⋊5S3, C3×D4⋊2S3, D12.22D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C4○D8, S32, S3×D4, C2×C3⋊D4, C2×S32, D8⋊3S3, Q8.13D6, S3×C3⋊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⋊D4C4○D8S32S3×D4C2×S32D8⋊3S3Q8.13D6S3×C3⋊D4D12.22D6
kernelD12.22D6S3×C3⋊C8Dic6⋊S3C32⋊3Q16C3×D4⋊S3C32⋊9SD16D12⋊5S3C3×D4⋊2S3D4⋊S3D4⋊2S3C3×Dic3S3×C6C3⋊C8Dic6C4×S3D12C3×D4Dic3D6C32D4C6C4C3C3C2C1
# 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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