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G = Dic6.22D6  order 288 = 25·32

9th non-split extension by Dic6 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial

Aliases: Dic6.22D6, C3⋊C8.10D6, (S3×Q8)⋊6S3, Q8.12S32, C3⋊Q16⋊4S3, (C4×S3).12D6, (S3×C6).38D4, C6.160(S3×D4), (C3×Q8).44D6, C3⋊6(Q16⋊S3), D6.Dic3⋊8C2, C32⋊2Q16⋊9C2, C32⋊11SD16⋊3C2, D6.16(C3⋊D4), C12.25(C22×S3), (C3×C12).25C23, (C3×Dic3).18D4, D6.6D6.2C2, C32⋊5SD16⋊10C2, C3⋊3(Q8.11D6), (S3×C12).24C22, C32⋊14(C8.C22), C12⋊S3.13C22, Dic3.13(C3⋊D4), (Q8×C32).7C22, C32⋊4C8.13C22, (C3×Dic6).20C22, (C3×S3×Q8)⋊3C2, C4.25(C2×S32), (C3×C3⋊Q16)⋊7C2, C2.34(S3×C3⋊D4), C6.56(C2×C3⋊D4), (C3×C6).140(C2×D4), (C3×C3⋊C8).16C22, SmallGroup(288,596)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — Dic6.22D6
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — D6.6D6 — Dic6.22D6
Lower central
C32 — C3×C6 — C3×C12 — Dic6.22D6
Upper central
C1 — C2 — C4 — Q8

Generators and relations for Dic6.22D6
 G = < a,b,c,d | a12=1, b2=c6=d2=a6, bab-1=a-1, cac-1=dad-1=a7, cbc-1=a9b, dbd-1=a3b, dcd-1=c5 >

Subgroups: 522 in 130 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, D4, Q8, Q8, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×Q8, C3×Q8, C8.C22, C3×Dic3, C3×Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C8⋊S3, C24⋊C2, C4.Dic3, Q8⋊2S3, C3⋊Q16, C3⋊Q16, C3×Q16, C4○D12, S3×Q8, Q8⋊3S3, C6×Q8, C3×C3⋊C8, C32⋊4C8, C6.D6, C3⋊D12, C3×Dic6, C3×Dic6, S3×C12, S3×C12, C12⋊S3, Q8×C32, Q16⋊S3, Q8.11D6, D6.Dic3, C32⋊5SD16, C32⋊2Q16, C3×C3⋊Q16, C32⋊11SD16, D6.6D6, C3×S3×Q8, Dic6.22D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C8.C22, S32, S3×D4, C2×C3⋊D4, C2×S32, Q16⋊S3, Q8.11D6, S3×C3⋊D4, Dic6.22D6

Smallest permutation representation of Dic6.22D6
►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 26 7 32)(2 25 8 31)(3 36 9 30)(4 35 10 29)(5 34 11 28)(6 33 12 27)(13 40 19 46)(14 39 20 45)(15 38 21 44)(16 37 22 43)(17 48 23 42)(18 47 24 41)

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

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

33 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E6A6B6C6D6E8A8B12A12B12C12D12E12F12G12H12I12J12K12L24A24B
order12223334444466666881212121212121212121212122424
size11636224246121222466123644448888121212241212

33 irreducible representations

dim11111111222222222244444448
type++++++++++++++++-++++
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6D6C3⋊D4C3⋊D4C8.C22S32S3×D4C2×S32Q16⋊S3Q8.11D6S3×C3⋊D4Dic6.22D6
kernelDic6.22D6D6.Dic3C32⋊5SD16C32⋊2Q16C3×C3⋊Q16C32⋊11SD16D6.6D6C3×S3×Q8C3⋊Q16S3×Q8C3×Dic3S3×C6C3⋊C8Dic6C4×S3C3×Q8Dic3D6C32Q8C6C4C3C3C2C1
# reps11111111111112122211112221

Matrix representation of Dic6.22D6 ►in GL8(ℤ)

000-10000
001-10000
01000000
-11000000
0000001-1
00000010
0000-1100
0000-1000
,
00001000
00000100
00000010
00000001
-10000000
0-1000000
00-100000
000-10000
,
10-110000
01-100000
-11-100000
-100-10000
0000-11-10
0000-100-1
0000-101-1
00000-110
,
0000-100-1
0000-11-10
000001-10
000010-11
01-100000
10-110000
10010000
1-1100000

G:=sub<GL(8,Integers())| [0,0,0,-1,0,0,0,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,0,0,0,-1,0,0,0],[0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[1,0,-1,-1,0,0,0,0,0,1,1,0,0,0,0,0,-1,-1,-1,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,-1,-1,-1,0,0,0,0,0,1,0,0,-1,0,0,0,0,-1,0,1,1,0,0,0,0,0,-1,-1,0],[0,0,0,0,0,1,1,1,0,0,0,0,1,0,0,-1,0,0,0,0,-1,-1,0,1,0,0,0,0,0,1,1,0,-1,-1,0,1,0,0,0,0,0,1,1,0,0,0,0,0,0,-1,-1,-1,0,0,0,0,-1,0,0,1,0,0,0,0] >;
 

Dic6.22D6 in GAP, Magma, Sage, TeX 

{\rm Dic}_6._{22}D_6
 
% in TeX
 
G:=Group("Dic6.22D6");
 
// GroupNames label
 
G:=SmallGroup(288,596);
 
// by ID
 
G=gap.SmallGroup(288,596);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,135,100,346,185,80,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^12=1,b^2=c^6=d^2=a^6,b*a*b^-1=a^-1,c*a*c^-1=d*a*d^-1=a^7,c*b*c^-1=a^9*b,d*b*d^-1=a^3*b,d*c*d^-1=c^5>;
 
// generators/relations
 

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