metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2).31D10, C4.66(C2×D20), (C2×C4).50D20, C20.418(C2×D4), (C2×C20).174D4, C23.21(C4×D5), (C2×M4(2))⋊12D5, C20.46D4⋊15C2, C4.12D20⋊15C2, (C10×M4(2))⋊20C2, (C2×C20).417C23, (C22×C4).140D10, C4.56(D10⋊C4), C20.114(C22⋊C4), (C2×D20).259C22, C4.Dic5.42C22, C22.7(D10⋊C4), (C22×C20).189C22, (C5×M4(2)).34C22, (C2×Dic10).286C22, C5⋊5(M4(2).8C22), (C2×C4×D5).4C4, C22.21(C2×C4×D5), (C2×C4).160(C4×D5), (C2×C5⋊D4).22C4, C4.111(C2×C5⋊D4), (C2×C20).282(C2×C4), (C2×C4○D20).12C2, (C2×C4).77(C5⋊D4), (C2×C4.Dic5)⋊16C2, (C2×Dic5).5(C2×C4), (C22×D5).6(C2×C4), C2.31(C2×D10⋊C4), C10.100(C2×C22⋊C4), (C2×C4).121(C22×D5), (C2×C10).116(C22×C4), (C22×C10).139(C2×C4), (C2×C10).130(C22⋊C4), SmallGroup(320,759)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2).31D10
G = < a,b,c,d | a8=b2=1, c10=d2=a4, bab=a5, ac=ca, dad-1=ab, bc=cb, bd=db, dcd-1=c9 >
Subgroups: 574 in 150 conjugacy classes, 59 normal (41 characteristic)
C1, C2, C2 [×5], C4 [×4], C4 [×2], C22 [×3], C22 [×5], C5, C8 [×4], C2×C4 [×6], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C23 [×2], D5 [×2], C10, C10 [×3], C2×C8 [×2], M4(2) [×2], M4(2) [×4], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×4], Dic5 [×2], C20 [×4], D10 [×4], C2×C10 [×3], C2×C10, C4.D4 [×2], C4.10D4 [×2], C2×M4(2), C2×M4(2), C2×C4○D4, C5⋊2C8 [×2], C40 [×2], Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×6], C22×D5 [×2], C22×C10, M4(2).8C22, C2×C5⋊2C8, C4.Dic5 [×2], C4.Dic5, C2×C40, C5×M4(2) [×2], C5×M4(2), C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×4], C2×C5⋊D4 [×2], C22×C20, C20.46D4 [×2], C4.12D20 [×2], C2×C4.Dic5, C10×M4(2), C2×C4○D20, M4(2).31D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C2×C22⋊C4, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, M4(2).8C22, D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, C2×D10⋊C4, M4(2).31D10
►On 80 points
(1 59 73 27 11 49 63 37)(2 60 74 28 12 50 64 38)(3 41 75 29 13 51 65 39)(4 42 76 30 14 52 66 40)(5 43 77 31 15 53 67 21)(6 44 78 32 16 54 68 22)(7 45 79 33 17 55 69 23)(8 46 80 34 18 56 70 24)(9 47 61 35 19 57 71 25)(10 48 62 36 20 58 72 26)
(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)
(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 10 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 22 31 32)(23 40 33 30)(24 29 34 39)(25 38 35 28)(26 27 36 37)(41 56 51 46)(42 45 52 55)(43 54 53 44)(47 50 57 60)(48 59 58 49)(61 64 71 74)(62 73 72 63)(65 80 75 70)(66 69 76 79)(67 78 77 68)
G:=sub<Sym(80)| (1,59,73,27,11,49,63,37)(2,60,74,28,12,50,64,38)(3,41,75,29,13,51,65,39)(4,42,76,30,14,52,66,40)(5,43,77,31,15,53,67,21)(6,44,78,32,16,54,68,22)(7,45,79,33,17,55,69,23)(8,46,80,34,18,56,70,24)(9,47,61,35,19,57,71,25)(10,48,62,36,20,58,72,26), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,22,31,32)(23,40,33,30)(24,29,34,39)(25,38,35,28)(26,27,36,37)(41,56,51,46)(42,45,52,55)(43,54,53,44)(47,50,57,60)(48,59,58,49)(61,64,71,74)(62,73,72,63)(65,80,75,70)(66,69,76,79)(67,78,77,68)>;
G:=Group( (1,59,73,27,11,49,63,37)(2,60,74,28,12,50,64,38)(3,41,75,29,13,51,65,39)(4,42,76,30,14,52,66,40)(5,43,77,31,15,53,67,21)(6,44,78,32,16,54,68,22)(7,45,79,33,17,55,69,23)(8,46,80,34,18,56,70,24)(9,47,61,35,19,57,71,25)(10,48,62,36,20,58,72,26), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,22,31,32)(23,40,33,30)(24,29,34,39)(25,38,35,28)(26,27,36,37)(41,56,51,46)(42,45,52,55)(43,54,53,44)(47,50,57,60)(48,59,58,49)(61,64,71,74)(62,73,72,63)(65,80,75,70)(66,69,76,79)(67,78,77,68) );
G=PermutationGroup([(1,59,73,27,11,49,63,37),(2,60,74,28,12,50,64,38),(3,41,75,29,13,51,65,39),(4,42,76,30,14,52,66,40),(5,43,77,31,15,53,67,21),(6,44,78,32,16,54,68,22),(7,45,79,33,17,55,69,23),(8,46,80,34,18,56,70,24),(9,47,61,35,19,57,71,25),(10,48,62,36,20,58,72,26)], [(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60)], [(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)], [(1,10,11,20),(2,19,12,9),(3,8,13,18),(4,17,14,7),(5,6,15,16),(21,22,31,32),(23,40,33,30),(24,29,34,39),(25,38,35,28),(26,27,36,37),(41,56,51,46),(42,45,52,55),(43,54,53,44),(47,50,57,60),(48,59,58,49),(61,64,71,74),(62,73,72,63),(65,80,75,70),(66,69,76,79),(67,78,77,68)])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 2 | 2 | 20 | 20 | 1 | 1 | 2 | 2 | 2 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D5 | D10 | D10 | C4×D5 | D20 | C5⋊D4 | C4×D5 | M4(2).8C22 | M4(2).31D10 |
| kernel | M4(2).31D10 | C20.46D4 | C4.12D20 | C2×C4.Dic5 | C10×M4(2) | C2×C4○D20 | C2×C4×D5 | C2×C5⋊D4 | C2×C20 | C2×M4(2) | M4(2) | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C5 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 4 | 2 | 4 | 2 | 4 | 8 | 8 | 4 | 2 | 8 |
Matrix representation of M4(2).31D10 ►in GL4(𝔽41) generated by
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 18 | 35 | 0 | 0 |
| 6 | 23 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 28 | 28 | 0 | 0 |
| 13 | 32 | 0 | 0 |
| 0 | 0 | 28 | 28 |
| 0 | 0 | 13 | 32 |
| 28 | 28 | 0 | 0 |
| 32 | 13 | 0 | 0 |
| 0 | 0 | 28 | 28 |
| 0 | 0 | 32 | 13 |
G:=sub<GL(4,GF(41))| [0,0,18,6,0,0,35,23,9,0,0,0,0,9,0,0],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[28,13,0,0,28,32,0,0,0,0,28,13,0,0,28,32],[28,32,0,0,28,13,0,0,0,0,28,32,0,0,28,13] >;
M4(2).31D10 in GAP, Magma, Sage, TeX
M_4(2)._{31}D_{10} % in TeX
G:=Group("M4(2).31D10"); // GroupNames label
G:=SmallGroup(320,759);
// by ID
G=gap.SmallGroup(320,759);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,422,58,1123,136,438,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=1,c^10=d^2=a^4,b*a*b=a^5,a*c=c*a,d*a*d^-1=a*b,b*c=c*b,b*d=d*b,d*c*d^-1=c^9>;
// generators/relations