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, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, D10, C2×C10, C2×C10, C4.D4, C4.10D4, C2×M4(2), C2×M4(2), C2×C4○D4, C5⋊2C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, M4(2).8C22, C2×C5⋊2C8, C4.Dic5, C4.Dic5, C2×C40, C5×M4(2), C5×M4(2), C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C22×C20, C20.46D4, C4.12D20, C2×C4.Dic5, C10×M4(2), C2×C4○D20, M4(2).31D10
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C2×C22⋊C4, C4×D5, D20, C5⋊D4, C22×D5, M4(2).8C22, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C2×D10⋊C4, M4(2).31D10
►On 80 points
(1 75 59 39 11 65 49 29)(2 76 60 40 12 66 50 30)(3 77 41 21 13 67 51 31)(4 78 42 22 14 68 52 32)(5 79 43 23 15 69 53 33)(6 80 44 24 16 70 54 34)(7 61 45 25 17 71 55 35)(8 62 46 26 18 72 56 36)(9 63 47 27 19 73 57 37)(10 64 48 28 20 74 58 38)
(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(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 26 31 36)(22 35 32 25)(23 24 33 34)(27 40 37 30)(28 29 38 39)(41 56 51 46)(42 45 52 55)(43 54 53 44)(47 50 57 60)(48 59 58 49)(61 68 71 78)(62 77 72 67)(63 66 73 76)(64 75 74 65)(69 80 79 70)
G:=sub<Sym(80)| (1,75,59,39,11,65,49,29)(2,76,60,40,12,66,50,30)(3,77,41,21,13,67,51,31)(4,78,42,22,14,68,52,32)(5,79,43,23,15,69,53,33)(6,80,44,24,16,70,54,34)(7,61,45,25,17,71,55,35)(8,62,46,26,18,72,56,36)(9,63,47,27,19,73,57,37)(10,64,48,28,20,74,58,38), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (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)(21,26,31,36)(22,35,32,25)(23,24,33,34)(27,40,37,30)(28,29,38,39)(41,56,51,46)(42,45,52,55)(43,54,53,44)(47,50,57,60)(48,59,58,49)(61,68,71,78)(62,77,72,67)(63,66,73,76)(64,75,74,65)(69,80,79,70)>;
G:=Group( (1,75,59,39,11,65,49,29)(2,76,60,40,12,66,50,30)(3,77,41,21,13,67,51,31)(4,78,42,22,14,68,52,32)(5,79,43,23,15,69,53,33)(6,80,44,24,16,70,54,34)(7,61,45,25,17,71,55,35)(8,62,46,26,18,72,56,36)(9,63,47,27,19,73,57,37)(10,64,48,28,20,74,58,38), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (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)(21,26,31,36)(22,35,32,25)(23,24,33,34)(27,40,37,30)(28,29,38,39)(41,56,51,46)(42,45,52,55)(43,54,53,44)(47,50,57,60)(48,59,58,49)(61,68,71,78)(62,77,72,67)(63,66,73,76)(64,75,74,65)(69,80,79,70) );
G=PermutationGroup([[(1,75,59,39,11,65,49,29),(2,76,60,40,12,66,50,30),(3,77,41,21,13,67,51,31),(4,78,42,22,14,68,52,32),(5,79,43,23,15,69,53,33),(6,80,44,24,16,70,54,34),(7,61,45,25,17,71,55,35),(8,62,46,26,18,72,56,36),(9,63,47,27,19,73,57,37),(10,64,48,28,20,74,58,38)], [(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80)], [(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),(21,26,31,36),(22,35,32,25),(23,24,33,34),(27,40,37,30),(28,29,38,39),(41,56,51,46),(42,45,52,55),(43,54,53,44),(47,50,57,60),(48,59,58,49),(61,68,71,78),(62,77,72,67),(63,66,73,76),(64,75,74,65),(69,80,79,70)]])
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