metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.8D20, C42⋊3D10, Q8.8D20, D20.33D4, M4(2)⋊4D10, Dic10.33D4, C4≀C2⋊2D5, (C5×D4).3D4, C20.4(C2×D4), (C5×Q8).3D4, C4○D4.2D10, C4.10(C2×D20), C4.126(D4×D5), D20⋊4C4⋊8C2, (C4×C20)⋊12C22, C8.D10⋊8C2, C5⋊2(D4.9D4), (C22×D5).3D4, C22.30(D4×D5), C10.28C22≀C2, D4⋊8D10.1C2, D4.9D10⋊1C2, C20.46D4⋊2C2, C4.D20⋊10C2, C4.Dic5⋊5C22, (C2×C20).265C23, C4○D20.14C22, (C2×D20).74C22, C2.31(C22⋊D20), (C2×Dic10)⋊14C22, (C5×M4(2))⋊11C22, (C5×C4≀C2)⋊2C2, (C2×C10).27(C2×D4), (C5×C4○D4).6C22, (C2×C4).110(C22×D5), SmallGroup(320,452)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)⋊D10
G = < a,b,c,d | a8=b2=c10=d2=1, bab=a5, cac-1=a-1b, dad=ab, cbc-1=a4b, bd=db, dcd=c-1 >
Subgroups: 782 in 152 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, C42, C22⋊C4, M4(2), M4(2), SD16, Q16, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C4.D4, C4≀C2, C4≀C2, C4.4D4, C8.C22, 2+ 1+4, C5⋊2C8, C40, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, C22×D5, D4.9D4, C40⋊C2, Dic20, C4.Dic5, D10⋊C4, D4.D5, C5⋊Q16, C4×C20, C5×M4(2), C2×Dic10, C2×D20, C2×D20, C4○D20, C4○D20, D4×D5, Q8⋊2D5, C5×C4○D4, D20⋊4C4, C20.46D4, C5×C4≀C2, C4.D20, C8.D10, D4.9D10, D4⋊8D10, M4(2)⋊D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, D20, C22×D5, D4.9D4, C2×D20, D4×D5, C22⋊D20, M4(2)⋊D10
►On 80 points
(1 75 16 51 14 49 10 65)(2 71 17 57 15 45 6 61)(3 77 18 53 11 41 7 67)(4 73 19 59 12 47 8 63)(5 79 20 55 13 43 9 69)(21 72 39 58 26 46 34 62)(22 42 40 68 27 78 35 54)(23 74 31 60 28 48 36 64)(24 44 32 70 29 80 37 56)(25 76 33 52 30 50 38 66)
(1 32)(2 38)(3 34)(4 40)(5 36)(6 30)(7 26)(8 22)(9 28)(10 24)(11 39)(12 35)(13 31)(14 37)(15 33)(16 29)(17 25)(18 21)(19 27)(20 23)(41 62)(42 59)(43 64)(44 51)(45 66)(46 53)(47 68)(48 55)(49 70)(50 57)(52 71)(54 73)(56 75)(58 77)(60 79)(61 76)(63 78)(65 80)(67 72)(69 74)
(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 14)(2 13)(3 12)(4 11)(5 15)(6 9)(7 8)(17 20)(18 19)(21 27)(22 26)(23 25)(28 30)(31 38)(32 37)(33 36)(34 35)(39 40)(41 54)(42 53)(43 52)(44 51)(45 60)(46 59)(47 58)(48 57)(49 56)(50 55)(61 74)(62 73)(63 72)(64 71)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)
G:=sub<Sym(80)| (1,75,16,51,14,49,10,65)(2,71,17,57,15,45,6,61)(3,77,18,53,11,41,7,67)(4,73,19,59,12,47,8,63)(5,79,20,55,13,43,9,69)(21,72,39,58,26,46,34,62)(22,42,40,68,27,78,35,54)(23,74,31,60,28,48,36,64)(24,44,32,70,29,80,37,56)(25,76,33,52,30,50,38,66), (1,32)(2,38)(3,34)(4,40)(5,36)(6,30)(7,26)(8,22)(9,28)(10,24)(11,39)(12,35)(13,31)(14,37)(15,33)(16,29)(17,25)(18,21)(19,27)(20,23)(41,62)(42,59)(43,64)(44,51)(45,66)(46,53)(47,68)(48,55)(49,70)(50,57)(52,71)(54,73)(56,75)(58,77)(60,79)(61,76)(63,78)(65,80)(67,72)(69,74), (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), (1,14)(2,13)(3,12)(4,11)(5,15)(6,9)(7,8)(17,20)(18,19)(21,27)(22,26)(23,25)(28,30)(31,38)(32,37)(33,36)(34,35)(39,40)(41,54)(42,53)(43,52)(44,51)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(61,74)(62,73)(63,72)(64,71)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)>;
G:=Group( (1,75,16,51,14,49,10,65)(2,71,17,57,15,45,6,61)(3,77,18,53,11,41,7,67)(4,73,19,59,12,47,8,63)(5,79,20,55,13,43,9,69)(21,72,39,58,26,46,34,62)(22,42,40,68,27,78,35,54)(23,74,31,60,28,48,36,64)(24,44,32,70,29,80,37,56)(25,76,33,52,30,50,38,66), (1,32)(2,38)(3,34)(4,40)(5,36)(6,30)(7,26)(8,22)(9,28)(10,24)(11,39)(12,35)(13,31)(14,37)(15,33)(16,29)(17,25)(18,21)(19,27)(20,23)(41,62)(42,59)(43,64)(44,51)(45,66)(46,53)(47,68)(48,55)(49,70)(50,57)(52,71)(54,73)(56,75)(58,77)(60,79)(61,76)(63,78)(65,80)(67,72)(69,74), (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), (1,14)(2,13)(3,12)(4,11)(5,15)(6,9)(7,8)(17,20)(18,19)(21,27)(22,26)(23,25)(28,30)(31,38)(32,37)(33,36)(34,35)(39,40)(41,54)(42,53)(43,52)(44,51)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(61,74)(62,73)(63,72)(64,71)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75) );
G=PermutationGroup([[(1,75,16,51,14,49,10,65),(2,71,17,57,15,45,6,61),(3,77,18,53,11,41,7,67),(4,73,19,59,12,47,8,63),(5,79,20,55,13,43,9,69),(21,72,39,58,26,46,34,62),(22,42,40,68,27,78,35,54),(23,74,31,60,28,48,36,64),(24,44,32,70,29,80,37,56),(25,76,33,52,30,50,38,66)], [(1,32),(2,38),(3,34),(4,40),(5,36),(6,30),(7,26),(8,22),(9,28),(10,24),(11,39),(12,35),(13,31),(14,37),(15,33),(16,29),(17,25),(18,21),(19,27),(20,23),(41,62),(42,59),(43,64),(44,51),(45,66),(46,53),(47,68),(48,55),(49,70),(50,57),(52,71),(54,73),(56,75),(58,77),(60,79),(61,76),(63,78),(65,80),(67,72),(69,74)], [(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)], [(1,14),(2,13),(3,12),(4,11),(5,15),(6,9),(7,8),(17,20),(18,19),(21,27),(22,26),(23,25),(28,30),(31,38),(32,37),(33,36),(34,35),(39,40),(41,54),(42,53),(43,52),(44,51),(45,60),(46,59),(47,58),(48,57),(49,56),(50,55),(61,74),(62,73),(63,72),(64,71),(65,80),(66,79),(67,78),(68,77),(69,76),(70,75)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20N | 20O | 20P | 40A | 40B | 40C | 40D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 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 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 20 | 40 | 2 | 2 | 8 | 40 | 2 | 2 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | D20 | D20 | D4.9D4 | D4×D5 | D4×D5 | M4(2)⋊D10 |
| kernel | M4(2)⋊D10 | D20⋊4C4 | C20.46D4 | C5×C4≀C2 | C4.D20 | C8.D10 | D4.9D10 | D4⋊8D10 | Dic10 | D20 | C5×D4 | C5×Q8 | C22×D5 | C4≀C2 | C42 | M4(2) | C4○D4 | D4 | Q8 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 8 |
Matrix representation of M4(2)⋊D10 ►in GL4(𝔽41) generated by
| 25 | 38 | 16 | 3 |
| 3 | 7 | 38 | 34 |
| 16 | 3 | 16 | 3 |
| 38 | 34 | 38 | 34 |
| 0 | 0 | 39 | 28 |
| 0 | 0 | 13 | 2 |
| 2 | 13 | 0 | 0 |
| 28 | 39 | 0 | 0 |
| 35 | 35 | 0 | 0 |
| 6 | 40 | 0 | 0 |
| 0 | 0 | 6 | 6 |
| 0 | 0 | 35 | 1 |
| 35 | 35 | 0 | 0 |
| 40 | 6 | 0 | 0 |
| 0 | 0 | 6 | 6 |
| 0 | 0 | 1 | 35 |
G:=sub<GL(4,GF(41))| [25,3,16,38,38,7,3,34,16,38,16,38,3,34,3,34],[0,0,2,28,0,0,13,39,39,13,0,0,28,2,0,0],[35,6,0,0,35,40,0,0,0,0,6,35,0,0,6,1],[35,40,0,0,35,6,0,0,0,0,6,1,0,0,6,35] >;
M4(2)⋊D10 in GAP, Magma, Sage, TeX
M_4(2)\rtimes D_{10} % in TeX
G:=Group("M4(2):D10"); // GroupNames label
G:=SmallGroup(320,452);
// by ID
G=gap.SmallGroup(320,452);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,58,1123,136,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^10=d^2=1,b*a*b=a^5,c*a*c^-1=a^-1*b,d*a*d=a*b,c*b*c^-1=a^4*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations