Copied to
clipboard

G = M4(2).22D6  order 192 = 26·3

5th non-split extension by M4(2) of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.24D6, M4(2).22D6, C4≀C2⋊6S3, D4⋊S3⋊3C4, C3⋊C8.34D4, D4.S3⋊3C4, C3⋊Q16⋊3C4, D4.3(C4×S3), C6.38(C4×D4), Q8.8(C4×S3), C3⋊2(C8.26D4), D12.C4⋊8C2, Q8⋊2S3⋊3C4, C4○D4.36D6, D12.6(C2×C4), C4.202(S3×D4), C42⋊4S3⋊6C2, C12.361(C2×D4), D4.Dic3⋊1C2, Dic6.6(C2×C4), C12.53D4⋊9C2, (C4×C12).50C22, C12.19(C22×C4), C42.S3⋊2C2, Q8.13D6.1C2, (C2×C12).263C23, C4○D12.12C22, C4.Dic3.8C22, C22.8(D4⋊2S3), C2.22(Dic3⋊4D4), (C3×M4(2)).24C22, C3⋊C8.2(C2×C4), C4.19(S3×C2×C4), (C3×C4≀C2)⋊11C2, (C3×D4).6(C2×C4), (C3×Q8).6(C2×C4), (C2×C3⋊C8).50C22, (C2×C6).34(C4○D4), (C3×C4○D4).4C22, (C2×C4).369(C22×S3), SmallGroup(192,382)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — M4(2).22D6
Chief series
C1 — C3 — C6 — C12 — C2×C12 — C4○D12 — Q8.13D6 — M4(2).22D6
Lower central
C3 — C6 — C12 — M4(2).22D6
Upper central
C1 — C4 — C2×C4 — C4≀C2

Generators and relations for M4(2).22D6
 G = < a,b,c,d | a8=b2=c6=1, d2=a6b, bab=a5, cac-1=dad-1=a-1b, cbc-1=a4b, bd=db, dcd-1=a6bc-1 >

Subgroups: 240 in 104 conjugacy classes, 45 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C2×C8, M4(2), M4(2), D8, SD16, Q16, C4○D4, C4○D4, C3⋊C8, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C8⋊C4, C4≀C2, C4≀C2, C8.C4, C8○D4, C4○D8, S3×C8, C8⋊S3, C2×C3⋊C8, C2×C3⋊C8, C4.Dic3, C4.Dic3, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C4×C12, C3×M4(2), C4○D12, C3×C4○D4, C8.26D4, C42.S3, C42⋊4S3, C12.53D4, C3×C4≀C2, D12.C4, D4.Dic3, Q8.13D6, M4(2).22D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, C22×S3, C4×D4, S3×C2×C4, S3×D4, D4⋊2S3, C8.26D4, Dic3⋊4D4, M4(2).22D6

Smallest permutation representation of M4(2).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 5)(3 7)(9 13)(11 15)(18 22)(20 24)(26 30)(28 32)(33 37)(35 39)(41 45)(43 47)

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

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

36 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E4F4G6A6B6C8A8B8C8D8E8F8G8H8I8J12A12B12C···12G12H24A24B
order12222344444446668888888888121212···12122424
size11241221124441224844666612121212224···4888

36 irreducible representations

dim111111111111222222224444
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C4C4C4C4S3D4D6D6D6C4○D4C4×S3C4×S3S3×D4D4⋊2S3C8.26D4M4(2).22D6
kernelM4(2).22D6C42.S3C42⋊4S3C12.53D4C3×C4≀C2D12.C4D4.Dic3Q8.13D6D4⋊S3D4.S3Q8⋊2S3C3⋊Q16C4≀C2C3⋊C8C42M4(2)C4○D4C2×C6D4Q8C4C22C3C1
# reps111111112222121112221124

Matrix representation of M4(2).22D6 ►in GL8(𝔽73)

00100000
00010000
720000000
072000000
000007200
000046000
000012001
0000012270
,
10000000
01000000
00100000
00010000
000072000
00000100
00000010
000000072
,
0072720000
00100000
7272000000
10000000
00000100
00001000
0000600046
0000059270
,
7118000000
202000000
002550000
0053710000
000039003
00000830
0000067650
0000130034

G:=sub<GL(8,GF(73))| [0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,46,12,0,0,0,0,0,72,0,0,12,0,0,0,0,0,0,0,27,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72],[0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,0,1,60,0,0,0,0,0,1,0,0,59,0,0,0,0,0,0,0,27,0,0,0,0,0,0,46,0],[71,20,0,0,0,0,0,0,18,2,0,0,0,0,0,0,0,0,2,53,0,0,0,0,0,0,55,71,0,0,0,0,0,0,0,0,39,0,0,13,0,0,0,0,0,8,67,0,0,0,0,0,0,3,65,0,0,0,0,0,3,0,0,34] >;
 

M4(2).22D6 in GAP, Magma, Sage, TeX 

M_4(2)._{22}D_6
 
% in TeX
 
G:=Group("M4(2).22D6");
 
// GroupNames label
 
G:=SmallGroup(192,382);
 
// by ID
 
G=gap.SmallGroup(192,382);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,219,58,136,1684,851,438,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^6=1,d^2=a^6*b,b*a*b=a^5,c*a*c^-1=d*a*d^-1=a^-1*b,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=a^6*b*c^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁