metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊11D10, Q16⋊10D10, D20.46D4, SD16⋊15D10, C20.17C24, C40.43C23, Dic10.46D4, D20.12C23, Dic10.11C23, C4○D8⋊5D5, C4○D4⋊2D10, (C2×C8)⋊14D10, C5⋊D4.2D4, D8⋊D5⋊6C2, D4⋊D5⋊4C22, (D5×SD16)⋊6C2, C5⋊3(D4○SD16), C4.144(D4×D5), Q8⋊D5⋊3C22, D4⋊D10⋊8C2, D4⋊8D10⋊6C2, (Q8×D5)⋊2C22, C22.9(D4×D5), (C2×C40)⋊17C22, Q16⋊D5⋊6C2, D10.53(C2×D4), C20.350(C2×D4), (C8×D5)⋊10C22, (C5×D8)⋊16C22, C5⋊2C8.8C23, D4.D5⋊3C22, (D4×D5).2C22, C5⋊Q16⋊2C22, C4.17(C23×D5), C8.17(C22×D5), SD16⋊3D5⋊6C2, D4.9D10⋊7C2, D4⋊2D5⋊2C22, C40⋊C2⋊21C22, C8⋊D5⋊16C22, Dic5.59(C2×D4), (C5×Q16)⋊14C22, D4.11(C22×D5), (C5×D4).11C23, (C4×D5).10C23, D4.10D10⋊5C2, D20.3C4⋊10C2, (C5×Q8).11C23, Q8.11(C22×D5), (C2×C20).534C23, C4○D20.55C22, (C5×SD16)⋊16C22, C10.118(C22×D4), C4.Dic5⋊31C22, Q8⋊2D5.2C22, (C2×Dic10)⋊38C22, (C2×D20).187C22, C2.91(C2×D4×D5), (C5×C4○D8)⋊7C2, (C2×C40⋊C2)⋊27C2, (C2×C10).14(C2×D4), (C5×C4○D4)⋊4C22, (C2×C4).233(C22×D5), SmallGroup(320,1442)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8⋊11D10
G = < a,b,c,d | a8=b2=c10=d2=1, bab=a-1, ac=ca, dad=a3, cbc-1=a4b, dbd=a6b, dcd=c-1 >
Subgroups: 1046 in 258 conjugacy classes, 99 normal (53 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, C2×C8, C2×C8, M4(2), D8, D8, SD16, SD16, Q16, Q16, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C8○D4, C2×SD16, C4○D8, C4○D8, C8⋊C22, C8.C22, 2+ 1+4, 2- 1+4, C5⋊2C8, C40, Dic10, Dic10, Dic10, C4×D5, C4×D5, D20, D20, D20, C2×Dic5, C5⋊D4, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, D4○SD16, C8×D5, C8⋊D5, C40⋊C2, C4.Dic5, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C2×C40, C5×D8, C5×SD16, C5×Q16, C2×Dic10, C2×Dic10, C2×D20, C2×D20, C4○D20, C4○D20, D4×D5, D4×D5, D4⋊2D5, D4⋊2D5, Q8×D5, Q8⋊2D5, C5×C4○D4, D20.3C4, C2×C40⋊C2, D8⋊D5, D5×SD16, SD16⋊3D5, Q16⋊D5, D4⋊D10, D4.9D10, C5×C4○D8, D4⋊8D10, D4.10D10, D8⋊11D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, C22×D5, D4○SD16, D4×D5, C23×D5, C2×D4×D5, D8⋊11D10
►On 80 points
(1 41 26 78 63 16 40 51)(2 42 27 79 64 17 31 52)(3 43 28 80 65 18 32 53)(4 44 29 71 66 19 33 54)(5 45 30 72 67 20 34 55)(6 46 21 73 68 11 35 56)(7 47 22 74 69 12 36 57)(8 48 23 75 70 13 37 58)(9 49 24 76 61 14 38 59)(10 50 25 77 62 15 39 60)
(1 51)(2 79)(3 53)(4 71)(5 55)(6 73)(7 57)(8 75)(9 59)(10 77)(11 35)(12 22)(13 37)(14 24)(15 39)(16 26)(17 31)(18 28)(19 33)(20 30)(21 46)(23 48)(25 50)(27 42)(29 44)(32 43)(34 45)(36 47)(38 49)(40 41)(52 64)(54 66)(56 68)(58 70)(60 62)(61 76)(63 78)(65 80)(67 72)(69 74)
(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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 50)(12 49)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(31 66)(32 65)(33 64)(34 63)(35 62)(36 61)(37 70)(38 69)(39 68)(40 67)(51 55)(52 54)(56 60)(57 59)(71 79)(72 78)(73 77)(74 76)
G:=sub<Sym(80)| (1,41,26,78,63,16,40,51)(2,42,27,79,64,17,31,52)(3,43,28,80,65,18,32,53)(4,44,29,71,66,19,33,54)(5,45,30,72,67,20,34,55)(6,46,21,73,68,11,35,56)(7,47,22,74,69,12,36,57)(8,48,23,75,70,13,37,58)(9,49,24,76,61,14,38,59)(10,50,25,77,62,15,39,60), (1,51)(2,79)(3,53)(4,71)(5,55)(6,73)(7,57)(8,75)(9,59)(10,77)(11,35)(12,22)(13,37)(14,24)(15,39)(16,26)(17,31)(18,28)(19,33)(20,30)(21,46)(23,48)(25,50)(27,42)(29,44)(32,43)(34,45)(36,47)(38,49)(40,41)(52,64)(54,66)(56,68)(58,70)(60,62)(61,76)(63,78)(65,80)(67,72)(69,74), (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,70)(38,69)(39,68)(40,67)(51,55)(52,54)(56,60)(57,59)(71,79)(72,78)(73,77)(74,76)>;
G:=Group( (1,41,26,78,63,16,40,51)(2,42,27,79,64,17,31,52)(3,43,28,80,65,18,32,53)(4,44,29,71,66,19,33,54)(5,45,30,72,67,20,34,55)(6,46,21,73,68,11,35,56)(7,47,22,74,69,12,36,57)(8,48,23,75,70,13,37,58)(9,49,24,76,61,14,38,59)(10,50,25,77,62,15,39,60), (1,51)(2,79)(3,53)(4,71)(5,55)(6,73)(7,57)(8,75)(9,59)(10,77)(11,35)(12,22)(13,37)(14,24)(15,39)(16,26)(17,31)(18,28)(19,33)(20,30)(21,46)(23,48)(25,50)(27,42)(29,44)(32,43)(34,45)(36,47)(38,49)(40,41)(52,64)(54,66)(56,68)(58,70)(60,62)(61,76)(63,78)(65,80)(67,72)(69,74), (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,70)(38,69)(39,68)(40,67)(51,55)(52,54)(56,60)(57,59)(71,79)(72,78)(73,77)(74,76) );
G=PermutationGroup([[(1,41,26,78,63,16,40,51),(2,42,27,79,64,17,31,52),(3,43,28,80,65,18,32,53),(4,44,29,71,66,19,33,54),(5,45,30,72,67,20,34,55),(6,46,21,73,68,11,35,56),(7,47,22,74,69,12,36,57),(8,48,23,75,70,13,37,58),(9,49,24,76,61,14,38,59),(10,50,25,77,62,15,39,60)], [(1,51),(2,79),(3,53),(4,71),(5,55),(6,73),(7,57),(8,75),(9,59),(10,77),(11,35),(12,22),(13,37),(14,24),(15,39),(16,26),(17,31),(18,28),(19,33),(20,30),(21,46),(23,48),(25,50),(27,42),(29,44),(32,43),(34,45),(36,47),(38,49),(40,41),(52,64),(54,66),(56,68),(58,70),(60,62),(61,76),(63,78),(65,80),(67,72),(69,74)], [(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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,50),(12,49),(13,48),(14,47),(15,46),(16,45),(17,44),(18,43),(19,42),(20,41),(31,66),(32,65),(33,64),(34,63),(35,62),(36,61),(37,70),(38,69),(39,68),(40,67),(51,55),(52,54),(56,60),(57,59),(71,79),(72,78),(73,77),(74,76)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | D10 | D10 | D4○SD16 | D4×D5 | D4×D5 | D8⋊11D10 |
| kernel | D8⋊11D10 | D20.3C4 | C2×C40⋊C2 | D8⋊D5 | D5×SD16 | SD16⋊3D5 | Q16⋊D5 | D4⋊D10 | D4.9D10 | C5×C4○D8 | D4⋊8D10 | D4.10D10 | Dic10 | D20 | C5⋊D4 | C4○D8 | C2×C8 | D8 | SD16 | Q16 | C4○D4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 2 | 2 | 2 | 8 |
Matrix representation of D8⋊11D10 ►in GL4(𝔽41) generated by
| 0 | 0 | 40 | 29 |
| 0 | 0 | 12 | 1 |
| 2 | 24 | 39 | 17 |
| 17 | 39 | 24 | 2 |
| 39 | 17 | 1 | 12 |
| 24 | 2 | 29 | 40 |
| 39 | 17 | 2 | 24 |
| 24 | 2 | 17 | 39 |
| 14 | 27 | 27 | 14 |
| 14 | 30 | 27 | 11 |
| 28 | 13 | 27 | 14 |
| 28 | 19 | 27 | 11 |
| 1 | 7 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 2 | 14 | 40 | 34 |
| 0 | 39 | 0 | 1 |
G:=sub<GL(4,GF(41))| [0,0,2,17,0,0,24,39,40,12,39,24,29,1,17,2],[39,24,39,24,17,2,17,2,1,29,2,17,12,40,24,39],[14,14,28,28,27,30,13,19,27,27,27,27,14,11,14,11],[1,0,2,0,7,40,14,39,0,0,40,0,0,0,34,1] >;
D8⋊11D10 in GAP, Magma, Sage, TeX
D_8\rtimes_{11}D_{10} % in TeX
G:=Group("D8:11D10"); // GroupNames label
G:=SmallGroup(320,1442);
// by ID
G=gap.SmallGroup(320,1442);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,387,570,185,136,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^10=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^3,c*b*c^-1=a^4*b,d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations