metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2)⋊26D6, C24.52C23, C12.70C24, (C2×C8)⋊22D6, C8○D12⋊15C2, C4○D12.5C4, C3⋊C8.32C23, D12.C4⋊12C2, (S3×C8)⋊11C22, (C2×C24)⋊30C22, (C2×D12).17C4, D12.29(C2×C4), C3⋊1(Q8○M4(2)), C6.33(C23×C4), C4.69(S3×C23), C23.27(C4×S3), C8.44(C22×S3), C8⋊S3⋊19C22, (C6×M4(2))⋊15C2, (C2×M4(2))⋊16S3, (S3×M4(2))⋊10C2, (C4×S3).36C23, C12.93(C22×C4), (C2×Dic6).17C4, Dic6.30(C2×C4), D6.14(C22×C4), (C22×C4).280D6, (C2×C12).510C23, C4○D12.59C22, C4.Dic3⋊40C22, (C3×M4(2))⋊31C22, Dic3.14(C22×C4), (C22×C12).265C22, C4.95(S3×C2×C4), (C2×C3⋊C8)⋊12C22, (C2×C4).58(C4×S3), C3⋊D4.4(C2×C4), C22.28(S3×C2×C4), C2.34(S3×C22×C4), (C4×S3).10(C2×C4), (C2×C3⋊D4).15C4, (C2×C12).132(C2×C4), (C2×C4○D12).22C2, (S3×C2×C4).153C22, (C2×C4.Dic3)⋊25C2, (C2×C6).26(C22×C4), (C22×C6).79(C2×C4), (C22×S3).27(C2×C4), (C2×C4).605(C22×S3), (C2×Dic3).37(C2×C4), SmallGroup(192,1304)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)⋊26D6
G = < a,b,c,d | a8=b2=c6=d2=1, bab=cac-1=dad=a5, bc=cb, dbd=a4b, dcd=c-1 >
Subgroups: 504 in 258 conjugacy classes, 147 normal (41 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C2×M4(2), C2×M4(2), C8○D4, C2×C4○D4, S3×C8, C8⋊S3, C2×C3⋊C8, C4.Dic3, C2×C24, C3×M4(2), C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, Q8○M4(2), C8○D12, S3×M4(2), D12.C4, C2×C4.Dic3, C6×M4(2), C2×C4○D12, M4(2)⋊26D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C24, C4×S3, C22×S3, C23×C4, S3×C2×C4, S3×C23, Q8○M4(2), S3×C22×C4, M4(2)⋊26D6
►On 48 points
(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 15)(2 12)(3 9)(4 14)(5 11)(6 16)(7 13)(8 10)(17 40)(18 37)(19 34)(20 39)(21 36)(22 33)(23 38)(24 35)(25 47)(26 44)(27 41)(28 46)(29 43)(30 48)(31 45)(32 42)
(1 32 40)(2 29 33 6 25 37)(3 26 34)(4 31 35 8 27 39)(5 28 36)(7 30 38)(9 44 19)(10 41 20 14 45 24)(11 46 21)(12 43 22 16 47 18)(13 48 23)(15 42 17)
(1 40)(2 37)(3 34)(4 39)(5 36)(6 33)(7 38)(8 35)(9 23)(10 20)(11 17)(12 22)(13 19)(14 24)(15 21)(16 18)(25 29)(27 31)(42 46)(44 48)
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), (1,15)(2,12)(3,9)(4,14)(5,11)(6,16)(7,13)(8,10)(17,40)(18,37)(19,34)(20,39)(21,36)(22,33)(23,38)(24,35)(25,47)(26,44)(27,41)(28,46)(29,43)(30,48)(31,45)(32,42), (1,32,40)(2,29,33,6,25,37)(3,26,34)(4,31,35,8,27,39)(5,28,36)(7,30,38)(9,44,19)(10,41,20,14,45,24)(11,46,21)(12,43,22,16,47,18)(13,48,23)(15,42,17), (1,40)(2,37)(3,34)(4,39)(5,36)(6,33)(7,38)(8,35)(9,23)(10,20)(11,17)(12,22)(13,19)(14,24)(15,21)(16,18)(25,29)(27,31)(42,46)(44,48)>;
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), (1,15)(2,12)(3,9)(4,14)(5,11)(6,16)(7,13)(8,10)(17,40)(18,37)(19,34)(20,39)(21,36)(22,33)(23,38)(24,35)(25,47)(26,44)(27,41)(28,46)(29,43)(30,48)(31,45)(32,42), (1,32,40)(2,29,33,6,25,37)(3,26,34)(4,31,35,8,27,39)(5,28,36)(7,30,38)(9,44,19)(10,41,20,14,45,24)(11,46,21)(12,43,22,16,47,18)(13,48,23)(15,42,17), (1,40)(2,37)(3,34)(4,39)(5,36)(6,33)(7,38)(8,35)(9,23)(10,20)(11,17)(12,22)(13,19)(14,24)(15,21)(16,18)(25,29)(27,31)(42,46)(44,48) );
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)], [(1,15),(2,12),(3,9),(4,14),(5,11),(6,16),(7,13),(8,10),(17,40),(18,37),(19,34),(20,39),(21,36),(22,33),(23,38),(24,35),(25,47),(26,44),(27,41),(28,46),(29,43),(30,48),(31,45),(32,42)], [(1,32,40),(2,29,33,6,25,37),(3,26,34),(4,31,35,8,27,39),(5,28,36),(7,30,38),(9,44,19),(10,41,20,14,45,24),(11,46,21),(12,43,22,16,47,18),(13,48,23),(15,42,17)], [(1,40),(2,37),(3,34),(4,39),(5,36),(6,33),(7,38),(8,35),(9,23),(10,20),(11,17),(12,22),(13,19),(14,24),(15,21),(16,18),(25,29),(27,31),(42,46),(44,48)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 8A | ··· | 8H | 8I | ··· | 8P | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D6 | D6 | D6 | C4×S3 | C4×S3 | Q8○M4(2) | M4(2)⋊26D6 |
| kernel | M4(2)⋊26D6 | C8○D12 | S3×M4(2) | D12.C4 | C2×C4.Dic3 | C6×M4(2) | C2×C4○D12 | C2×Dic6 | C2×D12 | C4○D12 | C2×C3⋊D4 | C2×M4(2) | C2×C8 | M4(2) | C22×C4 | C2×C4 | C23 | C3 | C1 |
| # reps | 1 | 4 | 4 | 4 | 1 | 1 | 1 | 2 | 2 | 8 | 4 | 1 | 2 | 4 | 1 | 6 | 2 | 2 | 4 |
Matrix representation of M4(2)⋊26D6 ►in GL4(𝔽73) generated by
| 47 | 0 | 2 | 0 |
| 1 | 46 | 0 | 2 |
| 50 | 0 | 26 | 0 |
| 63 | 60 | 72 | 27 |
| 30 | 60 | 0 | 0 |
| 13 | 43 | 0 | 0 |
| 20 | 57 | 43 | 13 |
| 16 | 36 | 60 | 30 |
| 72 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 11 | 10 | 1 | 1 |
| 27 | 37 | 72 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 11 | 10 | 1 | 1 |
| 0 | 64 | 0 | 72 |
G:=sub<GL(4,GF(73))| [47,1,50,63,0,46,0,60,2,0,26,72,0,2,0,27],[30,13,20,16,60,43,57,36,0,0,43,60,0,0,13,30],[72,1,11,27,72,0,10,37,0,0,1,72,0,0,1,0],[72,0,11,0,72,1,10,64,0,0,1,0,0,0,1,72] >;
M4(2)⋊26D6 in GAP, Magma, Sage, TeX
M_4(2)\rtimes_{26}D_6 % in TeX
G:=Group("M4(2):26D6"); // GroupNames label
G:=SmallGroup(192,1304);
// by ID
G=gap.SmallGroup(192,1304);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,570,80,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^6=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^5,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations