Copied to
clipboard

G = Dic3⋊5D12  order 288 = 25·32

2nd semidirect product of Dic3 and D12 acting through Inn(Dic3)

metabelian, supersoluble, monomial

Aliases: Dic3⋊5D12, C62.64C23, C12⋊6(C4×S3), C3⋊1(C4×D12), C32⋊6(C4×D4), C2.4(S3×D12), C6.18(S3×D4), C4⋊Dic3⋊17S3, (C3×Dic3)⋊9D4, (C4×Dic3)⋊6S3, C6.19(C2×D12), C12⋊S3⋊10C4, (C2×C12).281D6, C4⋊1(C6.D6), C3⋊1(Dic3⋊5D4), (Dic3×C12)⋊11C2, C6.12(C4○D12), (C2×Dic3).71D6, C6.D12⋊13C2, (C6×C12).102C22, C6.13(Q8⋊3S3), C2.4(D6.6D6), (C6×Dic3).64C22, (C2×C4).80S32, C6.33(S3×C2×C4), (C3×C12)⋊6(C2×C4), C22.36(C2×S32), (C3×C6).51(C2×D4), (C3×C4⋊Dic3)⋊13C2, (C2×C6.D6)⋊9C2, (C3×C6).38(C4○D4), (C3×C6).57(C22×C4), (C2×C6).83(C22×S3), C2.10(C2×C6.D6), (C2×C12⋊S3).13C2, (C22×C3⋊S3).17C22, (C2×C3⋊S3)⋊2(C2×C4), SmallGroup(288,542)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — Dic3⋊5D12
Chief series
C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C2×C6.D6 — Dic3⋊5D12
Lower central
C32 — C3×C6 — Dic3⋊5D12
Upper central
C1 — C22 — C2×C4

Generators and relations for Dic3⋊5D12
 G = < a,b,c,d | a6=c12=d2=1, b2=a3, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >

Subgroups: 962 in 215 conjugacy classes, 64 normal (34 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3⋊S3, C3×C6, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C4×D4, C3×Dic3, C3×Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, C4×Dic3, C4⋊Dic3, D6⋊C4, C4×C12, C3×C4⋊C4, S3×C2×C4, C2×D12, C6.D6, C6×Dic3, C6×Dic3, C12⋊S3, C6×C12, C22×C3⋊S3, C4×D12, Dic3⋊5D4, C6.D12, Dic3×C12, C3×C4⋊Dic3, C2×C6.D6, C2×C12⋊S3, Dic3⋊5D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, D12, C22×S3, C4×D4, S32, S3×C2×C4, C2×D12, C4○D12, S3×D4, Q8⋊3S3, C6.D6, C2×S32, C4×D12, Dic3⋊5D4, D6.6D6, S3×D12, C2×C6.D6, Dic3⋊5D12

Smallest permutation representation of Dic3⋊5D12
►On 48 points
Generators in S48
(1 24 9 20 5 16)(2 13 10 21 6 17)(3 14 11 22 7 18)(4 15 12 23 8 19)(25 47 29 39 33 43)(26 48 30 40 34 44)(27 37 31 41 35 45)(28 38 32 42 36 46)

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

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

54 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F4G···4L6A···6F6G6H6I12A12B12C12D12E···12J12K···12R12S12T12U12V
order122222223334444444···46···66661212121212···1212···1212121212
size1111181818182242233336···62···244422224···46···612121212

54 irreducible representations

dim11111112222222224444444
type+++++++++++++++++++
imageC1C2C2C2C2C2C4S3S3D4D6D6C4○D4D12C4×S3C4○D12S32S3×D4Q8⋊3S3C6.D6C2×S32D6.6D6S3×D12
kernelDic3⋊5D12C6.D12Dic3×C12C3×C4⋊Dic3C2×C6.D6C2×C12⋊S3C12⋊S3C4×Dic3C4⋊Dic3C3×Dic3C2×Dic3C2×C12C3×C6Dic3C12C6C2×C4C6C6C4C22C2C2
# reps12112181124224841112122

Matrix representation of Dic3⋊5D12 ►in GL6(𝔽13)

010000
1210000
0012000
0001200
0000120
0000012
,
050000
500000
008000
000800
000050
000005
,
100000
010000
009300
003400
0000012
000011
,
010000
100000
001000
0071200
00001212
000001

G:=sub<GL(6,GF(13))| [0,12,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,5,0,0,0,0,5,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,3,0,0,0,0,3,4,0,0,0,0,0,0,0,1,0,0,0,0,12,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,7,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,12,1] >;
 

Dic3⋊5D12 in GAP, Magma, Sage, TeX 

{\rm Dic}_3\rtimes_5D_{12}
 
% in TeX
 
G:=Group("Dic3:5D12");
 
// GroupNames label
 
G:=SmallGroup(288,542);
 
// by ID
 
G=gap.SmallGroup(288,542);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,64,219,100,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=c^12=d^2=1,b^2=a^3,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁