metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.24D10, M4(2).22D10, C4≀C2⋊6D5, D4⋊D5⋊7C4, Q8⋊D5⋊7C4, D4.D5⋊7C4, D4.3(C4×D5), C5⋊Q16⋊7C4, Q8.3(C4×D5), C5⋊5(C8.26D4), C10.68(C4×D4), C4.202(D4×D5), C5⋊2C8.50D4, D20⋊4C4⋊6C2, C4○D4.20D10, D20.20(C2×C4), C20.361(C2×D4), D4.Dic5⋊1C2, D20.2C4⋊8C2, C20.53D4⋊5C2, C20.54(C22×C4), (C4×C20).50C22, C42.D5⋊2C2, (C2×C20).263C23, Dic10.21(C2×C4), D4.8D10.1C2, C4○D20.12C22, C4.Dic5.8C22, C22.8(D4⋊2D5), C2.22(Dic5⋊4D4), (C5×M4(2)).16C22, (C5×C4≀C2)⋊7C2, C4.19(C2×C4×D5), C5⋊2C8.2(C2×C4), (C5×D4).20(C2×C4), (C5×Q8).21(C2×C4), (C5×C4○D4).4C22, (C2×C10).34(C4○D4), (C2×C5⋊2C8).50C22, (C2×C4).369(C22×D5), SmallGroup(320,450)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2).22D10
G = < a,b,c,d | a8=b2=c10=1, d2=a6b, bab=a5, cac-1=dad-1=a-1b, cbc-1=a4b, bd=db, dcd-1=a6bc-1 >
Subgroups: 326 in 104 conjugacy classes, 45 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, D5, C10, C10, C42, C2×C8, M4(2), M4(2), D8, SD16, Q16, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C8⋊C4, C4≀C2, C4≀C2, C8.C4, C8○D4, C4○D8, C5⋊2C8, C5⋊2C8, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C8.26D4, C8×D5, C8⋊D5, C2×C5⋊2C8, C2×C5⋊2C8, C4.Dic5, C4.Dic5, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C4×C20, C5×M4(2), C4○D20, C5×C4○D4, C42.D5, D20⋊4C4, C20.53D4, C5×C4≀C2, D20.2C4, D4.Dic5, D4.8D10, M4(2).22D10
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, D10, C4×D4, C4×D5, C22×D5, C8.26D4, C2×C4×D5, D4×D5, D4⋊2D5, Dic5⋊4D4, M4(2).22D10
►On 80 points
(1 51 15 6 33 20 56 38)(2 39 16 52 34 7 57 11)(3 53 17 8 35 12 58 40)(4 31 18 54 36 9 59 13)(5 55 19 10 37 14 60 32)(21 46 41 62 67 73 78 26)(22 27 42 47 68 63 79 74)(23 48 43 64 69 75 80 28)(24 29 44 49 70 65 71 76)(25 50 45 66 61 77 72 30)
(1 33)(3 35)(5 37)(7 39)(9 31)(11 52)(13 54)(15 56)(17 58)(19 60)(21 67)(23 69)(25 61)(27 63)(29 65)(41 78)(43 80)(45 72)(47 74)(49 76)
(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 49 50)(51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80)
(1 65 15 76 33 29 56 49)(2 48 57 28 34 75 16 64)(3 63 17 74 35 27 58 47)(4 46 59 26 36 73 18 62)(5 61 19 72 37 25 60 45)(6 44 51 24 38 71 20 70)(7 69 11 80 39 23 52 43)(8 42 53 22 40 79 12 68)(9 67 13 78 31 21 54 41)(10 50 55 30 32 77 14 66)
G:=sub<Sym(80)| (1,51,15,6,33,20,56,38)(2,39,16,52,34,7,57,11)(3,53,17,8,35,12,58,40)(4,31,18,54,36,9,59,13)(5,55,19,10,37,14,60,32)(21,46,41,62,67,73,78,26)(22,27,42,47,68,63,79,74)(23,48,43,64,69,75,80,28)(24,29,44,49,70,65,71,76)(25,50,45,66,61,77,72,30), (1,33)(3,35)(5,37)(7,39)(9,31)(11,52)(13,54)(15,56)(17,58)(19,60)(21,67)(23,69)(25,61)(27,63)(29,65)(41,78)(43,80)(45,72)(47,74)(49,76), (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)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80), (1,65,15,76,33,29,56,49)(2,48,57,28,34,75,16,64)(3,63,17,74,35,27,58,47)(4,46,59,26,36,73,18,62)(5,61,19,72,37,25,60,45)(6,44,51,24,38,71,20,70)(7,69,11,80,39,23,52,43)(8,42,53,22,40,79,12,68)(9,67,13,78,31,21,54,41)(10,50,55,30,32,77,14,66)>;
G:=Group( (1,51,15,6,33,20,56,38)(2,39,16,52,34,7,57,11)(3,53,17,8,35,12,58,40)(4,31,18,54,36,9,59,13)(5,55,19,10,37,14,60,32)(21,46,41,62,67,73,78,26)(22,27,42,47,68,63,79,74)(23,48,43,64,69,75,80,28)(24,29,44,49,70,65,71,76)(25,50,45,66,61,77,72,30), (1,33)(3,35)(5,37)(7,39)(9,31)(11,52)(13,54)(15,56)(17,58)(19,60)(21,67)(23,69)(25,61)(27,63)(29,65)(41,78)(43,80)(45,72)(47,74)(49,76), (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)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80), (1,65,15,76,33,29,56,49)(2,48,57,28,34,75,16,64)(3,63,17,74,35,27,58,47)(4,46,59,26,36,73,18,62)(5,61,19,72,37,25,60,45)(6,44,51,24,38,71,20,70)(7,69,11,80,39,23,52,43)(8,42,53,22,40,79,12,68)(9,67,13,78,31,21,54,41)(10,50,55,30,32,77,14,66) );
G=PermutationGroup([[(1,51,15,6,33,20,56,38),(2,39,16,52,34,7,57,11),(3,53,17,8,35,12,58,40),(4,31,18,54,36,9,59,13),(5,55,19,10,37,14,60,32),(21,46,41,62,67,73,78,26),(22,27,42,47,68,63,79,74),(23,48,43,64,69,75,80,28),(24,29,44,49,70,65,71,76),(25,50,45,66,61,77,72,30)], [(1,33),(3,35),(5,37),(7,39),(9,31),(11,52),(13,54),(15,56),(17,58),(19,60),(21,67),(23,69),(25,61),(27,63),(29,65),(41,78),(43,80),(45,72),(47,74),(49,76)], [(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),(41,42,43,44,45,46,47,48,49,50),(51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80)], [(1,65,15,76,33,29,56,49),(2,48,57,28,34,75,16,64),(3,63,17,74,35,27,58,47),(4,46,59,26,36,73,18,62),(5,61,19,72,37,25,60,45),(6,44,51,24,38,71,20,70),(7,69,11,80,39,23,52,43),(8,42,53,22,40,79,12,68),(9,67,13,78,31,21,54,41),(10,50,55,30,32,77,14,66)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10A | 10B | 10C | 10D | 10E | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20N | 20O | 20P | 40A | 40B | 40C | 40D |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 20 | 20 | 40 | 40 | 40 | 40 |
| size | 1 | 1 | 2 | 4 | 20 | 1 | 1 | 2 | 4 | 4 | 4 | 20 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C4×D5 | C4×D5 | C8.26D4 | D4×D5 | D4⋊2D5 | M4(2).22D10 |
| kernel | M4(2).22D10 | C42.D5 | D20⋊4C4 | C20.53D4 | C5×C4≀C2 | D20.2C4 | D4.Dic5 | D4.8D10 | D4⋊D5 | D4.D5 | Q8⋊D5 | C5⋊Q16 | C5⋊2C8 | C4≀C2 | C2×C10 | C42 | M4(2) | C4○D4 | D4 | Q8 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 8 |
Matrix representation of M4(2).22D10 ►in GL6(𝔽41)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 17 | 0 | 0 |
| 0 | 0 | 27 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 24 |
| 0 | 0 | 0 | 0 | 14 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 17 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 24 | 40 |
| 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 24 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 30 |
| 0 | 0 | 0 | 0 | 11 | 32 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 31 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 24 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 9 | 30 | 0 | 0 |
| 0 | 0 | 11 | 32 | 0 | 0 |
G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,27,0,0,0,0,17,40,0,0,0,0,0,0,40,14,0,0,0,0,24,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,17,0,0,0,0,0,1,0,0,0,0,0,0,1,24,0,0,0,0,0,40],[10,0,0,0,0,0,0,37,0,0,0,0,0,0,40,0,0,0,0,0,24,1,0,0,0,0,0,0,9,11,0,0,0,0,30,32],[0,31,0,0,0,0,4,0,0,0,0,0,0,0,0,0,9,11,0,0,0,0,30,32,0,0,40,0,0,0,0,0,24,1,0,0] >;
M4(2).22D10 in GAP, Magma, Sage, TeX
M_4(2)._{22}D_{10} % in TeX
G:=Group("M4(2).22D10"); // GroupNames label
G:=SmallGroup(320,450);
// by ID
G=gap.SmallGroup(320,450);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,219,58,136,1684,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^10=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