metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2).21D10, (C2×D20).7C4, (C4×D5).92D4, C20.96(C2×D4), C4.151(D4×D5), C4.10D4⋊6D5, (D5×M4(2))⋊7C2, (C2×C20).8C23, (C2×Q8).95D10, C20.10D4⋊4C2, C20.46D4⋊7C2, (Q8×C10).6C22, (C2×D20).44C22, C4.Dic5.5C22, D10.22(C22⋊C4), Dic5.55(C22⋊C4), (C5×M4(2)).13C22, C5⋊4(M4(2).8C22), (C2×C4×D5).3C4, (C2×C4).7(C4×D5), (C2×C4×D5).8C22, C22.17(C2×C4×D5), (C2×C20).21(C2×C4), C2.16(D5×C22⋊C4), (C2×C4).8(C22×D5), (C5×C4.10D4)⋊6C2, C10.56(C2×C22⋊C4), (C2×Q8⋊2D5).1C2, (C22×D5).3(C2×C4), (C2×C10).112(C22×C4), (C2×Dic5).141(C2×C4), SmallGroup(320,378)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2).21D10
G = < a,b,c,d | a8=b2=d2=1, c10=a4, bab=a5, cac-1=ab, dad=a5b, bc=cb, bd=db, dcd=a4c9 >
Subgroups: 622 in 150 conjugacy classes, 51 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, D10, C2×C10, C4.D4, C4.10D4, C4.10D4, C2×M4(2), C2×C4○D4, C5⋊2C8, C40, C4×D5, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C22×D5, M4(2).8C22, C8×D5, C8⋊D5, C4.Dic5, C5×M4(2), C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, Q8⋊2D5, Q8×C10, C20.46D4, C20.10D4, C5×C4.10D4, D5×M4(2), C2×Q8⋊2D5, M4(2).21D10
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).21D10
►On 80 points
(1 36 68 56 11 26 78 46)(2 27 79 57 12 37 69 47)(3 38 70 58 13 28 80 48)(4 29 61 59 14 39 71 49)(5 40 72 60 15 30 62 50)(6 31 63 41 16 21 73 51)(7 22 74 42 17 32 64 52)(8 33 65 43 18 23 75 53)(9 24 76 44 19 34 66 54)(10 35 67 45 20 25 77 55)
(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)(2 9)(3 8)(4 7)(5 6)(11 20)(12 19)(13 18)(14 17)(15 16)(21 30)(22 29)(23 28)(24 27)(25 26)(31 40)(32 39)(33 38)(34 37)(35 36)(41 50)(42 49)(43 48)(44 47)(45 46)(51 60)(52 59)(53 58)(54 57)(55 56)(61 64)(62 63)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)
G:=sub<Sym(80)| (1,36,68,56,11,26,78,46)(2,27,79,57,12,37,69,47)(3,38,70,58,13,28,80,48)(4,29,61,59,14,39,71,49)(5,40,72,60,15,30,62,50)(6,31,63,41,16,21,73,51)(7,22,74,42,17,32,64,52)(8,33,65,43,18,23,75,53)(9,24,76,44,19,34,66,54)(10,35,67,45,20,25,77,55), (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)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(21,30)(22,29)(23,28)(24,27)(25,26)(31,40)(32,39)(33,38)(34,37)(35,36)(41,50)(42,49)(43,48)(44,47)(45,46)(51,60)(52,59)(53,58)(54,57)(55,56)(61,64)(62,63)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)>;
G:=Group( (1,36,68,56,11,26,78,46)(2,27,79,57,12,37,69,47)(3,38,70,58,13,28,80,48)(4,29,61,59,14,39,71,49)(5,40,72,60,15,30,62,50)(6,31,63,41,16,21,73,51)(7,22,74,42,17,32,64,52)(8,33,65,43,18,23,75,53)(9,24,76,44,19,34,66,54)(10,35,67,45,20,25,77,55), (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)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(21,30)(22,29)(23,28)(24,27)(25,26)(31,40)(32,39)(33,38)(34,37)(35,36)(41,50)(42,49)(43,48)(44,47)(45,46)(51,60)(52,59)(53,58)(54,57)(55,56)(61,64)(62,63)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73) );
G=PermutationGroup([[(1,36,68,56,11,26,78,46),(2,27,79,57,12,37,69,47),(3,38,70,58,13,28,80,48),(4,29,61,59,14,39,71,49),(5,40,72,60,15,30,62,50),(6,31,63,41,16,21,73,51),(7,22,74,42,17,32,64,52),(8,33,65,43,18,23,75,53),(9,24,76,44,19,34,66,54),(10,35,67,45,20,25,77,55)], [(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),(2,9),(3,8),(4,7),(5,6),(11,20),(12,19),(13,18),(14,17),(15,16),(21,30),(22,29),(23,28),(24,27),(25,26),(31,40),(32,39),(33,38),(34,37),(35,36),(41,50),(42,49),(43,48),(44,47),(45,46),(51,60),(52,59),(53,58),(54,57),(55,56),(61,64),(62,63),(65,80),(66,79),(67,78),(68,77),(69,76),(70,75),(71,74),(72,73)]])
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 | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 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 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 5 | 5 | 10 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 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 | 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).21D10 |
| kernel | M4(2).21D10 | C20.46D4 | C20.10D4 | C5×C4.10D4 | D5×M4(2) | C2×Q8⋊2D5 | C2×C4×D5 | C2×D20 | C4×D5 | C4.10D4 | M4(2) | C2×Q8 | C2×C4 | 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).21D10 ►in GL8(𝔽41)
| 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 32 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 31 | 36 | 39 |
| 0 | 0 | 0 | 0 | 32 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 8 | 13 | 32 | 21 |
| 0 | 0 | 0 | 0 | 20 | 25 | 16 | 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 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 36 | 40 |
| 40 | 2 | 7 | 27 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 34 | 0 | 0 | 0 | 0 |
| 34 | 14 | 7 | 27 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 37 | 0 |
| 0 | 0 | 0 | 0 | 9 | 31 | 36 | 39 |
| 0 | 0 | 0 | 0 | 21 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 21 | 30 | 25 | 10 |
| 1 | 39 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 27 | 40 | 2 | 0 | 0 | 0 | 0 |
| 0 | 34 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 32 | 10 | 5 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 29 | 12 | 16 | 31 |
G:=sub<GL(8,GF(41))| [9,9,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,9,9,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,9,32,8,20,0,0,0,0,31,0,13,25,0,0,0,0,36,5,32,16,0,0,0,0,39,0,21,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,9,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,40],[40,0,34,0,0,0,0,0,2,1,14,7,0,0,0,0,7,0,7,0,0,0,0,0,27,34,27,34,0,0,0,0,0,0,0,0,40,9,21,21,0,0,0,0,0,31,0,30,0,0,0,0,37,36,1,25,0,0,0,0,0,39,0,10],[1,0,7,0,0,0,0,0,39,40,27,34,0,0,0,0,0,0,40,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,32,0,29,0,0,0,0,0,10,0,12,0,0,0,0,4,5,40,16,0,0,0,0,0,2,0,31] >;
M4(2).21D10 in GAP, Magma, Sage, TeX
M_4(2)._{21}D_{10} % in TeX
G:=Group("M4(2).21D10"); // GroupNames label
G:=SmallGroup(320,378);
// by ID
G=gap.SmallGroup(320,378);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,219,58,570,136,438,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=d^2=1,c^10=a^4,b*a*b=a^5,c*a*c^-1=a*b,d*a*d=a^5*b,b*c=c*b,b*d=d*b,d*c*d=a^4*c^9>;
// generators/relations