Copied to
clipboard

G = M4(2).21D6order 192 = 26·3

4th non-split extension by M4(2) of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M4(2).21D6, (C4×S3).36D4, (C2×D12).4C4, C4.151(S3×D4), C12.96(C2×D4), C4.10D46S3, (S3×M4(2))⋊7C2, (C2×C12).8C23, (C2×Q8).119D6, (C6×Q8).6C22, C12.10D44C2, D6.3(C22⋊C4), C12.46D411C2, (C2×D12).40C22, C4.Dic3.5C22, Dic3.17(C22⋊C4), (C3×M4(2)).21C22, C32(M4(2).8C22), (S3×C2×C4).3C4, (C2×C4).7(C4×S3), (C2×C12).7(C2×C4), (S3×C2×C4).4C22, C22.17(S3×C2×C4), C2.16(S3×C22⋊C4), C6.15(C2×C22⋊C4), (C2×C4).8(C22×S3), (C22×S3).3(C2×C4), (C2×C6).11(C22×C4), (C2×Q83S3).1C2, (C3×C4.10D4)⋊10C2, (C2×Dic3).86(C2×C4), SmallGroup(192,310)

Series: Derived Chief Lower central Upper central

C1C2×C6 — M4(2).21D6
C1C3C6C12C2×C12S3×C2×C4C2×Q83S3 — M4(2).21D6
C3C6C2×C6 — M4(2).21D6
C1C2C2×C4C4.10D4

Generators and relations for M4(2).21D6
 G = < a,b,c,d | a8=b2=d2=1, c6=a4, bab=a5, cac-1=ab, dad=a5b, bc=cb, bd=db, dcd=a4c5 >

Subgroups: 432 in 150 conjugacy classes, 51 normal (21 characteristic)
C1, C2, C2 [×5], C3, C4 [×2], C4 [×4], C22, C22 [×7], S3 [×4], C6, C6, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×9], D4 [×6], Q8 [×2], C23 [×3], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], D6 [×5], C2×C6, C2×C8 [×2], M4(2) [×2], M4(2) [×4], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×4], C3⋊C8 [×2], C24 [×2], C4×S3 [×4], C4×S3 [×4], D12 [×6], C2×Dic3, C2×C12, C2×C12 [×2], C3×Q8 [×2], C22×S3, C22×S3 [×2], C4.D4 [×2], C4.10D4, C4.10D4, C2×M4(2) [×2], C2×C4○D4, S3×C8 [×2], C8⋊S3 [×2], C4.Dic3 [×2], C3×M4(2) [×2], S3×C2×C4, S3×C2×C4 [×2], C2×D12, C2×D12 [×2], Q83S3 [×4], C6×Q8, M4(2).8C22, C12.46D4 [×2], C12.10D4, C3×C4.10D4, S3×M4(2) [×2], C2×Q83S3, M4(2).21D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], C22×S3, C2×C22⋊C4, S3×C2×C4, S3×D4 [×2], M4(2).8C22, S3×C22⋊C4, M4(2).21D6

Smallest permutation representation of M4(2).21D6
On 48 points
Generators in S48
(1 26 43 15 7 32 37 21)(2 33 38 16 8 27 44 22)(3 28 45 17 9 34 39 23)(4 35 40 18 10 29 46 24)(5 30 47 19 11 36 41 13)(6 25 42 20 12 31 48 14)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(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 6)(2 5)(3 4)(7 12)(8 11)(9 10)(13 16)(14 15)(17 24)(18 23)(19 22)(20 21)(25 26)(27 36)(28 35)(29 34)(30 33)(31 32)(37 42)(38 41)(39 40)(43 48)(44 47)(45 46)

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

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

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

33 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G6A6B8A8B8C8D8E8F8G8H12A12B12C12D24A24B24C24D
order12222223444444466888888881212121224242424
size112661212222334462444441212121244888888

33 irreducible representations

dim1111111122222448
type++++++++++++
imageC1C2C2C2C2C2C4C4S3D4D6D6C4×S3S3×D4M4(2).8C22M4(2).21D6
kernelM4(2).21D6C12.46D4C12.10D4C3×C4.10D4S3×M4(2)C2×Q83S3S3×C2×C4C2×D12C4.10D4C4×S3M4(2)C2×Q8C2×C4C4C3C1
# reps1211214414214221

Matrix representation of M4(2).21D6 in GL6(𝔽73)

100000
010000
0018181871
0051478
0023262770
0026402527
,
100000
010000
001000
00657200
000010
001901872
,
0720000
1720000
0010480
00424603
0030720
007205827
,
1720000
0720000
00720250
003127070
000010
001481546

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,5,23,26,0,0,18,1,26,40,0,0,18,47,27,25,0,0,71,8,70,27],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,65,0,19,0,0,0,72,0,0,0,0,0,0,1,18,0,0,0,0,0,72],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,42,3,72,0,0,0,46,0,0,0,0,48,0,72,58,0,0,0,3,0,27],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,72,31,0,1,0,0,0,27,0,48,0,0,25,0,1,15,0,0,0,70,0,46] >;

M4(2).21D6 in GAP, Magma, Sage, TeX

M_4(2)._{21}D_6
% in TeX

G:=Group("M4(2).21D6");
// GroupNames label

G:=SmallGroup(192,310);
// by ID

G=gap.SmallGroup(192,310);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,232,219,58,570,136,438,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=d^2=1,c^6=a^4,b*a*b=a^5,c*a*c^-1=a*b,d*a*d=a^5*b,b*c=c*b,b*d=d*b,d*c*d=a^4*c^5>;
// generators/relations

׿
×
𝔽