Copied to
clipboard

G = Dic6.19D6  order 288 = 25·32

6th non-split extension by Dic6 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial

Aliases: Dic6.19D6, D4.2S32, C3⋊C8.5D6, D4⋊2S3.S3, D4.S3⋊1S3, (S3×C6).9D4, (C3×D4).8D6, (C4×S3).6D6, (S3×Dic6)⋊5C2, C6.149(S3×D4), D6.7(C3⋊D4), C3⋊6(D4.D6), D6.Dic3⋊1C2, (C3×C12).6C23, C12.6(C22×S3), C32⋊2Q16⋊6C2, C3⋊2(Q8.14D6), C32⋊9SD16⋊2C2, (C3×Dic3).35D4, C32⋊3Q16⋊11C2, C32⋊9(C8.C22), (S3×C12).13C22, C32⋊4C8.6C22, (D4×C32).2C22, Dic3.16(C3⋊D4), C32⋊4Q8.6C22, (C3×Dic6).10C22, C4.6(C2×S32), (C3×D4.S3)⋊1C2, C2.23(S3×C3⋊D4), C6.45(C2×C3⋊D4), (C3×C6).121(C2×D4), (C3×C3⋊C8).10C22, (C3×D4⋊2S3).1C2, SmallGroup(288,577)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — Dic6.19D6
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — S3×Dic6 — Dic6.19D6
Lower central
C32 — C3×C6 — C3×C12 — Dic6.19D6
Upper central
C1 — C2 — C4 — D4

Generators and relations for Dic6.19D6
 G = < a,b,c,d | a12=c6=1, b2=d2=a6, bab-1=a-1, cac-1=dad-1=a7, cbc-1=dbd-1=a9b, dcd-1=a6c-1 >

Subgroups: 458 in 130 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, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, C8.C22, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C62, C8⋊S3, Dic12, C4.Dic3, D4.S3, D4.S3, C3⋊Q16, C3×SD16, C2×Dic6, D4⋊2S3, S3×Q8, C3×C4○D4, C3×C3⋊C8, C32⋊4C8, S3×Dic3, C32⋊2Q8, C3×Dic6, S3×C12, C6×Dic3, C3×C3⋊D4, C32⋊4Q8, D4×C32, D4.D6, Q8.14D6, D6.Dic3, C32⋊2Q16, C32⋊3Q16, C3×D4.S3, C32⋊9SD16, S3×Dic6, C3×D4⋊2S3, Dic6.19D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C8.C22, S32, S3×D4, C2×C3⋊D4, C2×S32, D4.D6, Q8.14D6, S3×C3⋊D4, Dic6.19D6

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

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

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

33 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E6A6B6C6D6E6F6G6H6I8A8B12A12B12C12D12E12F12G12H24A24B
order1222333444446666666668812121212121212122424
size11462242612123622444888121236446681212241212

33 irreducible representations

dim11111111222222222244444448
type++++++++++++++++-+++---
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6D6C3⋊D4C3⋊D4C8.C22S32S3×D4C2×S32D4.D6Q8.14D6S3×C3⋊D4Dic6.19D6
kernelDic6.19D6D6.Dic3C32⋊2Q16C32⋊3Q16C3×D4.S3C32⋊9SD16S3×Dic6C3×D4⋊2S3D4.S3D4⋊2S3C3×Dic3S3×C6C3⋊C8Dic6C4×S3C3×D4Dic3D6C32D4C6C4C3C3C2C1
# reps11111111111112122211112221

Matrix representation of Dic6.19D6 ►in GL8(𝔽73)

640000000
448000000
00800000
0029640000
000016600
0000427200
000000727
000000311
,
00100000
00010000
10000000
01000000
00000010
00000001
000072000
000007200
,
640000000
448000000
006400000
004480000
00001000
0000427200
000000727
00000001
,
0036670000
009370000
3667000000
937000000
00004550652
0000028530
0000021450
00002066528

G:=sub<GL(8,GF(73))| [64,44,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,29,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,42,0,0,0,0,0,0,66,72,0,0,0,0,0,0,0,0,72,31,0,0,0,0,0,0,7,1],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[64,44,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64,44,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,42,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,7,1],[0,0,36,9,0,0,0,0,0,0,67,37,0,0,0,0,36,9,0,0,0,0,0,0,67,37,0,0,0,0,0,0,0,0,0,0,45,0,0,20,0,0,0,0,50,28,21,6,0,0,0,0,6,53,45,65,0,0,0,0,52,0,0,28] >;
 

Dic6.19D6 in GAP, Magma, Sage, TeX 

{\rm Dic}_6._{19}D_6
 
% in TeX
 
G:=Group("Dic6.19D6");
 
// GroupNames label
 
G:=SmallGroup(288,577);
 
// by ID
 
G=gap.SmallGroup(288,577);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,422,135,346,185,80,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^12=c^6=1,b^2=d^2=a^6,b*a*b^-1=a^-1,c*a*c^-1=d*a*d^-1=a^7,c*b*c^-1=d*b*d^-1=a^9*b,d*c*d^-1=a^6*c^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁