metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2).19D10, (C4×D5).91D4, C4.147(D4×D5), C20.90(C2×D4), C4.D4⋊6D5, C23.7(C4×D5), C20.D4⋊4C2, (D5×M4(2))⋊6C2, (C2×C20).2C23, (C2×D4).122D10, C4.12D20⋊5C2, (D4×C10).12C22, (C22×Dic5).4C4, C4.Dic5.1C22, D10.21(C22⋊C4), (C5×M4(2)).9C22, Dic5.54(C22⋊C4), (C2×Dic10).48C22, C5⋊3(M4(2).8C22), (C2×C5⋊D4).2C4, (C2×C4×D5).6C22, C22.15(C2×C4×D5), (C5×C4.D4)⋊6C2, C2.14(D5×C22⋊C4), (C2×C4).2(C22×D5), (C2×D4⋊2D5).2C2, C10.54(C2×C22⋊C4), (C22×C10).7(C2×C4), (C2×Dic5).2(C2×C4), (C22×D5).20(C2×C4), (C2×C10).110(C22×C4), SmallGroup(320,372)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2).19D10
G = < a,b,c,d | a8=b2=c10=1, d2=a4, bab=a5, cac-1=ab, dad-1=a5b, bc=cb, bd=db, dcd-1=a4c-1 >
Subgroups: 558 in 150 conjugacy classes, 51 normal (21 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, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C10, C4.D4, C4.D4, C4.10D4, C2×M4(2), C2×C4○D4, C5⋊2C8, C40, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, M4(2).8C22, C8×D5, C8⋊D5, C4.Dic5, C5×M4(2), C2×Dic10, C2×C4×D5, D4⋊2D5, C22×Dic5, C2×C5⋊D4, D4×C10, C4.12D20, C20.D4, C5×C4.D4, D5×M4(2), C2×D4⋊2D5, M4(2).19D10
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C2×C22⋊C4, C4×D5, C22×D5, M4(2).8C22, C2×C4×D5, D4×D5, D5×C22⋊C4, M4(2).19D10
►On 80 points
(1 41 40 78 63 11 57 29)(2 12 58 79 64 42 31 30)(3 43 32 80 65 13 59 21)(4 14 60 71 66 44 33 22)(5 45 34 72 67 15 51 23)(6 16 52 73 68 46 35 24)(7 47 36 74 69 17 53 25)(8 18 54 75 70 48 37 26)(9 49 38 76 61 19 55 27)(10 20 56 77 62 50 39 28)
(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 80)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)
(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 63 62)(2 61 64 9)(3 8 65 70)(4 69 66 7)(5 6 67 68)(11 50 41 20)(12 19 42 49)(13 48 43 18)(14 17 44 47)(15 46 45 16)(21 75 80 26)(22 25 71 74)(23 73 72 24)(27 79 76 30)(28 29 77 78)(31 55 58 38)(32 37 59 54)(33 53 60 36)(34 35 51 52)(39 57 56 40)
G:=sub<Sym(80)| (1,41,40,78,63,11,57,29)(2,12,58,79,64,42,31,30)(3,43,32,80,65,13,59,21)(4,14,60,71,66,44,33,22)(5,45,34,72,67,15,51,23)(6,16,52,73,68,46,35,24)(7,47,36,74,69,17,53,25)(8,18,54,75,70,48,37,26)(9,49,38,76,61,19,55,27)(10,20,56,77,62,50,39,28), (11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,80)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79), (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), (1,10,63,62)(2,61,64,9)(3,8,65,70)(4,69,66,7)(5,6,67,68)(11,50,41,20)(12,19,42,49)(13,48,43,18)(14,17,44,47)(15,46,45,16)(21,75,80,26)(22,25,71,74)(23,73,72,24)(27,79,76,30)(28,29,77,78)(31,55,58,38)(32,37,59,54)(33,53,60,36)(34,35,51,52)(39,57,56,40)>;
G:=Group( (1,41,40,78,63,11,57,29)(2,12,58,79,64,42,31,30)(3,43,32,80,65,13,59,21)(4,14,60,71,66,44,33,22)(5,45,34,72,67,15,51,23)(6,16,52,73,68,46,35,24)(7,47,36,74,69,17,53,25)(8,18,54,75,70,48,37,26)(9,49,38,76,61,19,55,27)(10,20,56,77,62,50,39,28), (11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,80)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79), (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), (1,10,63,62)(2,61,64,9)(3,8,65,70)(4,69,66,7)(5,6,67,68)(11,50,41,20)(12,19,42,49)(13,48,43,18)(14,17,44,47)(15,46,45,16)(21,75,80,26)(22,25,71,74)(23,73,72,24)(27,79,76,30)(28,29,77,78)(31,55,58,38)(32,37,59,54)(33,53,60,36)(34,35,51,52)(39,57,56,40) );
G=PermutationGroup([[(1,41,40,78,63,11,57,29),(2,12,58,79,64,42,31,30),(3,43,32,80,65,13,59,21),(4,14,60,71,66,44,33,22),(5,45,34,72,67,15,51,23),(6,16,52,73,68,46,35,24),(7,47,36,74,69,17,53,25),(8,18,54,75,70,48,37,26),(9,49,38,76,61,19,55,27),(10,20,56,77,62,50,39,28)], [(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,80),(22,71),(23,72),(24,73),(25,74),(26,75),(27,76),(28,77),(29,78),(30,79)], [(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)], [(1,10,63,62),(2,61,64,9),(3,8,65,70),(4,69,66,7),(5,6,67,68),(11,50,41,20),(12,19,42,49),(13,48,43,18),(14,17,44,47),(15,46,45,16),(21,75,80,26),(22,25,71,74),(23,73,72,24),(27,79,76,30),(28,29,77,78),(31,55,58,38),(32,37,59,54),(33,53,60,36),(34,35,51,52),(39,57,56,40)]])
44 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 | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| 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 | 10 | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 4 | 4 | 10 | 10 | 2 | 2 | 5 | 5 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D5 | D10 | D10 | C4×D5 | M4(2).8C22 | D4×D5 | M4(2).19D10 |
| kernel | M4(2).19D10 | C4.12D20 | C20.D4 | C5×C4.D4 | D5×M4(2) | C2×D4⋊2D5 | C22×Dic5 | C2×C5⋊D4 | C4×D5 | C4.D4 | M4(2) | C2×D4 | C23 | C5 | C4 | C1 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 4 | 4 | 2 | 4 | 2 | 8 | 2 | 4 | 2 |
Matrix representation of M4(2).19D10 ►in GL8(𝔽41)
| 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 6 | 25 | 39 |
| 0 | 0 | 0 | 0 | 7 | 3 | 23 | 40 |
| 0 | 0 | 0 | 0 | 12 | 20 | 29 | 19 |
| 0 | 0 | 0 | 0 | 12 | 10 | 9 | 10 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 25 | 40 |
| 0 | 6 | 0 | 6 | 0 | 0 | 0 | 0 |
| 6 | 0 | 6 | 0 | 0 | 0 | 0 | 0 |
| 0 | 35 | 0 | 1 | 0 | 0 | 0 | 0 |
| 35 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 | 21 | 0 |
| 0 | 0 | 0 | 0 | 12 | 14 | 21 | 9 |
| 0 | 0 | 0 | 0 | 3 | 0 | 26 | 0 |
| 0 | 0 | 0 | 0 | 9 | 33 | 29 | 27 |
| 0 | 35 | 0 | 35 | 0 | 0 | 0 | 0 |
| 35 | 0 | 35 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 6 | 0 | 0 | 0 | 0 |
| 40 | 0 | 6 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 0 | 20 | 0 |
| 0 | 0 | 0 | 0 | 29 | 27 | 20 | 32 |
| 0 | 0 | 0 | 0 | 1 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 9 | 31 | 12 | 14 |
G:=sub<GL(8,GF(41))| [9,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,40,7,12,12,0,0,0,0,6,3,20,10,0,0,0,0,25,23,29,9,0,0,0,0,39,40,19,10],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,1,0,2,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,25,0,0,0,0,0,0,0,40],[0,6,0,35,0,0,0,0,6,0,35,0,0,0,0,0,0,6,0,1,0,0,0,0,6,0,1,0,0,0,0,0,0,0,0,0,15,12,3,9,0,0,0,0,0,14,0,33,0,0,0,0,21,21,26,29,0,0,0,0,0,9,0,27],[0,35,0,40,0,0,0,0,35,0,40,0,0,0,0,0,0,35,0,6,0,0,0,0,35,0,6,0,0,0,0,0,0,0,0,0,26,29,1,9,0,0,0,0,0,27,0,31,0,0,0,0,20,20,15,12,0,0,0,0,0,32,0,14] >;
M4(2).19D10 in GAP, Magma, Sage, TeX
M_4(2)._{19}D_{10} % in TeX
G:=Group("M4(2).19D10"); // GroupNames label
G:=SmallGroup(320,372);
// by ID
G=gap.SmallGroup(320,372);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,219,58,570,136,438,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^10=1,d^2=a^4,b*a*b=a^5,c*a*c^-1=a*b,d*a*d^-1=a^5*b,b*c=c*b,b*d=d*b,d*c*d^-1=a^4*c^-1>;
// generators/relations