metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2).21D6, (C4×S3).36D4, (C2×D12).4C4, C4.151(S3×D4), C12.96(C2×D4), C4.10D4⋊6S3, (S3×M4(2))⋊7C2, (C2×C12).8C23, (C2×Q8).119D6, (C6×Q8).6C22, C12.10D4⋊4C2, D6.3(C22⋊C4), C12.46D4⋊11C2, (C2×D12).40C22, C4.Dic3.5C22, Dic3.17(C22⋊C4), (C3×M4(2)).21C22, C3⋊2(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×Q8⋊3S3).1C2, (C3×C4.10D4)⋊10C2, (C2×Dic3).86(C2×C4), SmallGroup(192,310)
Series: Derived ►Chief ►Lower central ►Upper central
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, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C3⋊C8, C24, C4×S3, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C22×S3, C4.D4, C4.10D4, C4.10D4, C2×M4(2), C2×C4○D4, S3×C8, C8⋊S3, C4.Dic3, C3×M4(2), S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, Q8⋊3S3, C6×Q8, M4(2).8C22, C12.46D4, C12.10D4, C3×C4.10D4, S3×M4(2), C2×Q8⋊3S3, M4(2).21D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, C22×S3, C2×C22⋊C4, S3×C2×C4, S3×D4, M4(2).8C22, S3×C22⋊C4, M4(2).21D6
►On 48 points
(1 37 30 23 7 43 36 17)(2 44 25 24 8 38 31 18)(3 39 32 13 9 45 26 19)(4 46 27 14 10 40 33 20)(5 41 34 15 11 47 28 21)(6 48 29 16 12 42 35 22)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(37 43)(38 44)(39 45)(40 46)(41 47)(42 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 6)(2 5)(3 4)(7 12)(8 11)(9 10)(13 20)(14 19)(15 18)(16 17)(21 24)(22 23)(25 28)(26 27)(29 36)(30 35)(31 34)(32 33)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)
G:=sub<Sym(48)| (1,37,30,23,7,43,36,17)(2,44,25,24,8,38,31,18)(3,39,32,13,9,45,26,19)(4,46,27,14,10,40,33,20)(5,41,34,15,11,47,28,21)(6,48,29,16,12,42,35,22), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (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), (1,6)(2,5)(3,4)(7,12)(8,11)(9,10)(13,20)(14,19)(15,18)(16,17)(21,24)(22,23)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)>;
G:=Group( (1,37,30,23,7,43,36,17)(2,44,25,24,8,38,31,18)(3,39,32,13,9,45,26,19)(4,46,27,14,10,40,33,20)(5,41,34,15,11,47,28,21)(6,48,29,16,12,42,35,22), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (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), (1,6)(2,5)(3,4)(7,12)(8,11)(9,10)(13,20)(14,19)(15,18)(16,17)(21,24)(22,23)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43) );
G=PermutationGroup([[(1,37,30,23,7,43,36,17),(2,44,25,24,8,38,31,18),(3,39,32,13,9,45,26,19),(4,46,27,14,10,40,33,20),(5,41,34,15,11,47,28,21),(6,48,29,16,12,42,35,22)], [(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(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)], [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10),(13,20),(14,19),(15,18),(16,17),(21,24),(22,23),(25,28),(26,27),(29,36),(30,35),(31,34),(32,33),(37,48),(38,47),(39,46),(40,45),(41,44),(42,43)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 6 | 2 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D6 | D6 | C4×S3 | S3×D4 | M4(2).8C22 | M4(2).21D6 |
| kernel | M4(2).21D6 | C12.46D4 | C12.10D4 | C3×C4.10D4 | S3×M4(2) | C2×Q8⋊3S3 | S3×C2×C4 | C2×D12 | C4.10D4 | C4×S3 | M4(2) | C2×Q8 | C2×C4 | C4 | C3 | C1 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 4 | 1 | 4 | 2 | 1 | 4 | 2 | 2 | 1 |
Matrix representation of M4(2).21D6 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 18 | 18 | 71 |
| 0 | 0 | 5 | 1 | 47 | 8 |
| 0 | 0 | 23 | 26 | 27 | 70 |
| 0 | 0 | 26 | 40 | 25 | 27 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 65 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 19 | 0 | 18 | 72 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 48 | 0 |
| 0 | 0 | 42 | 46 | 0 | 3 |
| 0 | 0 | 3 | 0 | 72 | 0 |
| 0 | 0 | 72 | 0 | 58 | 27 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 25 | 0 |
| 0 | 0 | 31 | 27 | 0 | 70 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 48 | 15 | 46 |
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