metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊7M4(2), (C2×C8)⋊18D10, C22⋊C8⋊11D5, C4.195(D4×D5), (C2×C40)⋊20C22, (C4×D5).102D4, C20.354(C2×D4), (C2×C10)⋊5M4(2), D10⋊1C8⋊17C2, (C23×D5).8C4, C23.47(C4×D5), C22⋊3(C8⋊D5), C5⋊4(C24.4C4), C2.12(D5×M4(2)), C20.55D4⋊23C2, (C2×C20).821C23, (C22×C4).304D10, C10.39(C2×M4(2)), D10.35(C22⋊C4), (C22×Dic5).16C4, Dic5.36(C22⋊C4), (C22×C20).338C22, (C2×C4×D5).22C4, C2.9(C2×C8⋊D5), C2.9(D5×C22⋊C4), (C2×C8⋊D5)⋊11C2, (C5×C22⋊C8)⋊15C2, (C2×C4).132(C4×D5), (D5×C22×C4).17C2, C22.103(C2×C4×D5), (C2×C20).327(C2×C4), (C2×C5⋊2C8)⋊27C22, C10.49(C2×C22⋊C4), (C2×C4×D5).343C22, (C22×D5).99(C2×C4), (C2×C4).763(C22×D5), (C22×C10).107(C2×C4), (C2×C10).177(C22×C4), (C2×Dic5).140(C2×C4), SmallGroup(320,353)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10⋊7M4(2)
G = < a,b,c,d | a10=b2=c8=d2=1, bab=cac-1=dad=a-1, cbc-1=a3b, dbd=a8b, dcd=c5 >
Subgroups: 734 in 190 conjugacy classes, 61 normal (33 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C23, C23, D5, C10, C10, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C24, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C22⋊C8, C22⋊C8, C2×M4(2), C23×C4, C5⋊2C8, C40, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C24.4C4, C8⋊D5, C2×C5⋊2C8, C2×C40, C2×C4×D5, C2×C4×D5, C22×Dic5, C22×C20, C23×D5, D10⋊1C8, C20.55D4, C5×C22⋊C8, C2×C8⋊D5, D5×C22×C4, D10⋊7M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, M4(2), C22×C4, C2×D4, D10, C2×C22⋊C4, C2×M4(2), C4×D5, C22×D5, C24.4C4, C8⋊D5, C2×C4×D5, D4×D5, D5×C22⋊C4, C2×C8⋊D5, D5×M4(2), D10⋊7M4(2)
►On 80 points
(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 5)(2 4)(6 10)(7 9)(11 20)(12 19)(13 18)(14 17)(15 16)(22 30)(23 29)(24 28)(25 27)(31 39)(32 38)(33 37)(34 36)(41 43)(44 50)(45 49)(46 48)(51 54)(52 53)(55 60)(56 59)(57 58)(61 68)(62 67)(63 66)(64 65)(69 70)(71 74)(72 73)(75 80)(76 79)(77 78)
(1 73 45 70 29 16 38 53)(2 72 46 69 30 15 39 52)(3 71 47 68 21 14 40 51)(4 80 48 67 22 13 31 60)(5 79 49 66 23 12 32 59)(6 78 50 65 24 11 33 58)(7 77 41 64 25 20 34 57)(8 76 42 63 26 19 35 56)(9 75 43 62 27 18 36 55)(10 74 44 61 28 17 37 54)
(1 24)(2 23)(3 22)(4 21)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 16)(12 15)(13 14)(17 20)(18 19)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 50)(39 49)(40 48)(51 60)(52 59)(53 58)(54 57)(55 56)(61 64)(62 63)(65 70)(66 69)(67 68)(71 80)(72 79)(73 78)(74 77)(75 76)
G:=sub<Sym(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), (1,5)(2,4)(6,10)(7,9)(11,20)(12,19)(13,18)(14,17)(15,16)(22,30)(23,29)(24,28)(25,27)(31,39)(32,38)(33,37)(34,36)(41,43)(44,50)(45,49)(46,48)(51,54)(52,53)(55,60)(56,59)(57,58)(61,68)(62,67)(63,66)(64,65)(69,70)(71,74)(72,73)(75,80)(76,79)(77,78), (1,73,45,70,29,16,38,53)(2,72,46,69,30,15,39,52)(3,71,47,68,21,14,40,51)(4,80,48,67,22,13,31,60)(5,79,49,66,23,12,32,59)(6,78,50,65,24,11,33,58)(7,77,41,64,25,20,34,57)(8,76,42,63,26,19,35,56)(9,75,43,62,27,18,36,55)(10,74,44,61,28,17,37,54), (1,24)(2,23)(3,22)(4,21)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,16)(12,15)(13,14)(17,20)(18,19)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,50)(39,49)(40,48)(51,60)(52,59)(53,58)(54,57)(55,56)(61,64)(62,63)(65,70)(66,69)(67,68)(71,80)(72,79)(73,78)(74,77)(75,76)>;
G:=Group( (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), (1,5)(2,4)(6,10)(7,9)(11,20)(12,19)(13,18)(14,17)(15,16)(22,30)(23,29)(24,28)(25,27)(31,39)(32,38)(33,37)(34,36)(41,43)(44,50)(45,49)(46,48)(51,54)(52,53)(55,60)(56,59)(57,58)(61,68)(62,67)(63,66)(64,65)(69,70)(71,74)(72,73)(75,80)(76,79)(77,78), (1,73,45,70,29,16,38,53)(2,72,46,69,30,15,39,52)(3,71,47,68,21,14,40,51)(4,80,48,67,22,13,31,60)(5,79,49,66,23,12,32,59)(6,78,50,65,24,11,33,58)(7,77,41,64,25,20,34,57)(8,76,42,63,26,19,35,56)(9,75,43,62,27,18,36,55)(10,74,44,61,28,17,37,54), (1,24)(2,23)(3,22)(4,21)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,16)(12,15)(13,14)(17,20)(18,19)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,50)(39,49)(40,48)(51,60)(52,59)(53,58)(54,57)(55,56)(61,64)(62,63)(65,70)(66,69)(67,68)(71,80)(72,79)(73,78)(74,77)(75,76) );
G=PermutationGroup([[(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)], [(1,5),(2,4),(6,10),(7,9),(11,20),(12,19),(13,18),(14,17),(15,16),(22,30),(23,29),(24,28),(25,27),(31,39),(32,38),(33,37),(34,36),(41,43),(44,50),(45,49),(46,48),(51,54),(52,53),(55,60),(56,59),(57,58),(61,68),(62,67),(63,66),(64,65),(69,70),(71,74),(72,73),(75,80),(76,79),(77,78)], [(1,73,45,70,29,16,38,53),(2,72,46,69,30,15,39,52),(3,71,47,68,21,14,40,51),(4,80,48,67,22,13,31,60),(5,79,49,66,23,12,32,59),(6,78,50,65,24,11,33,58),(7,77,41,64,25,20,34,57),(8,76,42,63,26,19,35,56),(9,75,43,62,27,18,36,55),(10,74,44,61,28,17,37,54)], [(1,24),(2,23),(3,22),(4,21),(5,30),(6,29),(7,28),(8,27),(9,26),(10,25),(11,16),(12,15),(13,14),(17,20),(18,19),(31,47),(32,46),(33,45),(34,44),(35,43),(36,42),(37,41),(38,50),(39,49),(40,48),(51,60),(52,59),(53,58),(54,57),(55,56),(61,64),(62,63),(65,70),(66,69),(67,68),(71,80),(72,79),(73,78),(74,77),(75,76)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 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 | 2 | 2 | 2 | 4 | 4 | 4 | 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 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D5 | M4(2) | M4(2) | D10 | D10 | C4×D5 | C4×D5 | C8⋊D5 | D4×D5 | D5×M4(2) |
| kernel | D10⋊7M4(2) | D10⋊1C8 | C20.55D4 | C5×C22⋊C8 | C2×C8⋊D5 | D5×C22×C4 | C2×C4×D5 | C22×Dic5 | C23×D5 | C4×D5 | C22⋊C8 | D10 | C2×C10 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 2 | 4 | 4 | 16 | 4 | 4 |
Matrix representation of D10⋊7M4(2) ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 32 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 40 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 25 | 40 |
| 32 | 39 | 0 | 0 | 0 | 0 |
| 36 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 22 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 37 |
| 0 | 0 | 0 | 0 | 20 | 32 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 32 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,7,34,0,0,0,0,7,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,32,0,0,0,0,0,40,0,0,0,0,0,0,7,40,0,0,0,0,7,34,0,0,0,0,0,0,1,25,0,0,0,0,0,40],[32,36,0,0,0,0,39,9,0,0,0,0,0,0,32,22,0,0,0,0,0,9,0,0,0,0,0,0,9,20,0,0,0,0,37,32],[1,32,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;
D10⋊7M4(2) in GAP, Magma, Sage, TeX
D_{10}\rtimes_7M_4(2) % in TeX
G:=Group("D10:7M4(2)"); // GroupNames label
G:=SmallGroup(320,353);
// by ID
G=gap.SmallGroup(320,353);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,219,58,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^8=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^3*b,d*b*d=a^8*b,d*c*d=c^5>;
// generators/relations