metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.10D10, Q8.11D10, C5⋊22- 1+4, C10.13C24, C20.27C23, D10.8C23, D20.14C22, Dic5.8C23, Dic10.14C22, C4○D4⋊4D5, (Q8×D5)⋊5C2, C4○D20⋊9C2, C5⋊D4.C22, D4⋊2D5⋊5C2, (C2×C4).25D10, (C2×C10).5C23, (C4×D5).6C22, C2.14(C23×D5), C4.34(C22×D5), (C2×Dic10)⋊14C2, (C2×C20).49C22, (C5×D4).10C22, (C5×Q8).11C22, C22.4(C22×D5), (C2×Dic5).22C22, (C5×C4○D4)⋊5C2, SmallGroup(160,225)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.10D10
G = < a,b,c,d | a4=b2=1, c10=d2=a2, bab=cac-1=dad-1=a-1, cbc-1=a2b, bd=db, dcd-1=c9 >
Subgroups: 376 in 146 conjugacy classes, 85 normal (12 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C2×C4, C2×C4, D4, D4, Q8, Q8, D5, C10, C10, C2×Q8, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, 2- 1+4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×Q8, C2×Dic10, C4○D20, D4⋊2D5, Q8×D5, C5×C4○D4, D4.10D10
Quotients: C1, C2, C22, C23, D5, C24, D10, 2- 1+4, C22×D5, C23×D5, D4.10D10
►On 80 points
(1 68 11 78)(2 79 12 69)(3 70 13 80)(4 61 14 71)(5 72 15 62)(6 63 16 73)(7 74 17 64)(8 65 18 75)(9 76 19 66)(10 67 20 77)(21 51 31 41)(22 42 32 52)(23 53 33 43)(24 44 34 54)(25 55 35 45)(26 46 36 56)(27 57 37 47)(28 48 38 58)(29 59 39 49)(30 50 40 60)
(1 23)(2 34)(3 25)(4 36)(5 27)(6 38)(7 29)(8 40)(9 31)(10 22)(11 33)(12 24)(13 35)(14 26)(15 37)(16 28)(17 39)(18 30)(19 21)(20 32)(41 66)(42 77)(43 68)(44 79)(45 70)(46 61)(47 72)(48 63)(49 74)(50 65)(51 76)(52 67)(53 78)(54 69)(55 80)(56 71)(57 62)(58 73)(59 64)(60 75)
(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 47 11 57)(2 56 12 46)(3 45 13 55)(4 54 14 44)(5 43 15 53)(6 52 16 42)(7 41 17 51)(8 50 18 60)(9 59 19 49)(10 48 20 58)(21 74 31 64)(22 63 32 73)(23 72 33 62)(24 61 34 71)(25 70 35 80)(26 79 36 69)(27 68 37 78)(28 77 38 67)(29 66 39 76)(30 75 40 65)
G:=sub<Sym(80)| (1,68,11,78)(2,79,12,69)(3,70,13,80)(4,61,14,71)(5,72,15,62)(6,63,16,73)(7,74,17,64)(8,65,18,75)(9,76,19,66)(10,67,20,77)(21,51,31,41)(22,42,32,52)(23,53,33,43)(24,44,34,54)(25,55,35,45)(26,46,36,56)(27,57,37,47)(28,48,38,58)(29,59,39,49)(30,50,40,60), (1,23)(2,34)(3,25)(4,36)(5,27)(6,38)(7,29)(8,40)(9,31)(10,22)(11,33)(12,24)(13,35)(14,26)(15,37)(16,28)(17,39)(18,30)(19,21)(20,32)(41,66)(42,77)(43,68)(44,79)(45,70)(46,61)(47,72)(48,63)(49,74)(50,65)(51,76)(52,67)(53,78)(54,69)(55,80)(56,71)(57,62)(58,73)(59,64)(60,75), (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), (1,47,11,57)(2,56,12,46)(3,45,13,55)(4,54,14,44)(5,43,15,53)(6,52,16,42)(7,41,17,51)(8,50,18,60)(9,59,19,49)(10,48,20,58)(21,74,31,64)(22,63,32,73)(23,72,33,62)(24,61,34,71)(25,70,35,80)(26,79,36,69)(27,68,37,78)(28,77,38,67)(29,66,39,76)(30,75,40,65)>;
G:=Group( (1,68,11,78)(2,79,12,69)(3,70,13,80)(4,61,14,71)(5,72,15,62)(6,63,16,73)(7,74,17,64)(8,65,18,75)(9,76,19,66)(10,67,20,77)(21,51,31,41)(22,42,32,52)(23,53,33,43)(24,44,34,54)(25,55,35,45)(26,46,36,56)(27,57,37,47)(28,48,38,58)(29,59,39,49)(30,50,40,60), (1,23)(2,34)(3,25)(4,36)(5,27)(6,38)(7,29)(8,40)(9,31)(10,22)(11,33)(12,24)(13,35)(14,26)(15,37)(16,28)(17,39)(18,30)(19,21)(20,32)(41,66)(42,77)(43,68)(44,79)(45,70)(46,61)(47,72)(48,63)(49,74)(50,65)(51,76)(52,67)(53,78)(54,69)(55,80)(56,71)(57,62)(58,73)(59,64)(60,75), (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), (1,47,11,57)(2,56,12,46)(3,45,13,55)(4,54,14,44)(5,43,15,53)(6,52,16,42)(7,41,17,51)(8,50,18,60)(9,59,19,49)(10,48,20,58)(21,74,31,64)(22,63,32,73)(23,72,33,62)(24,61,34,71)(25,70,35,80)(26,79,36,69)(27,68,37,78)(28,77,38,67)(29,66,39,76)(30,75,40,65) );
G=PermutationGroup([[(1,68,11,78),(2,79,12,69),(3,70,13,80),(4,61,14,71),(5,72,15,62),(6,63,16,73),(7,74,17,64),(8,65,18,75),(9,76,19,66),(10,67,20,77),(21,51,31,41),(22,42,32,52),(23,53,33,43),(24,44,34,54),(25,55,35,45),(26,46,36,56),(27,57,37,47),(28,48,38,58),(29,59,39,49),(30,50,40,60)], [(1,23),(2,34),(3,25),(4,36),(5,27),(6,38),(7,29),(8,40),(9,31),(10,22),(11,33),(12,24),(13,35),(14,26),(15,37),(16,28),(17,39),(18,30),(19,21),(20,32),(41,66),(42,77),(43,68),(44,79),(45,70),(46,61),(47,72),(48,63),(49,74),(50,65),(51,76),(52,67),(53,78),(54,69),(55,80),(56,71),(57,62),(58,73),(59,64),(60,75)], [(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)], [(1,47,11,57),(2,56,12,46),(3,45,13,55),(4,54,14,44),(5,43,15,53),(6,52,16,42),(7,41,17,51),(8,50,18,60),(9,59,19,49),(10,48,20,58),(21,74,31,64),(22,63,32,73),(23,72,33,62),(24,61,34,71),(25,70,35,80),(26,79,36,69),(27,68,37,78),(28,77,38,67),(29,66,39,76),(30,75,40,65)]])
D4.10D10 is a maximal subgroup of
D4.9D20 D4.10D20 D20.38D4 D20.40D4 D4.11D20 D4.13D20 D8⋊11D10 D20.47D4 D8⋊6D10 D20.44D4 D20.33C23 D20.35C23 C10.C25 D20.37C23 D5×2- 1+4 D20.A4 D20.39D6 C30.C24 C15⋊2- 1+4 D12.29D10 D4.10D30
D4.10D10 is a maximal quotient of
C42.87D10 C42.89D10 C42.90D10 C42.91D10 C42.92D10 C42.94D10 C42.96D10 C42.98D10 C42.99D10 D4×Dic10 C42.105D10 C42.106D10 C42.108D10 D20⋊24D4 Dic10⋊23D4 D4⋊6D20 C42.115D10 C42.118D10 Dic10⋊10Q8 Q8⋊6Dic10 C42.125D10 Q8×D20 C42.134D10 C42.135D10 C10.682- 1+4 Dic10⋊19D4 C10.352+ 1+4 C10.362+ 1+4 C10.392+ 1+4 C10.732- 1+4 C10.742- 1+4 C10.502+ 1+4 C10.152- 1+4 C10.162- 1+4 Dic10⋊21D4 C10.772- 1+4 C10.572+ 1+4 C10.582+ 1+4 C10.792- 1+4 C10.802- 1+4 C10.812- 1+4 C10.822- 1+4 C10.632+ 1+4 C10.842- 1+4 C10.852- 1+4 C10.692+ 1+4 C42.137D10 C42.139D10 C42.140D10 C42.141D10 Dic10⋊10D4 C42.144D10 C42.145D10 Dic10⋊7Q8 C42.147D10 C42.148D10 C42.152D10 C42.154D10 C42.155D10 C42.157D10 C42.159D10 C42.160D10 C42.161D10 C42.162D10 C42.164D10 C42.165D10 C10.1042- 1+4 C10.1052- 1+4 C10.1062- 1+4 C10.1072- 1+4 C10.1472+ 1+4 D20.39D6 C30.C24 C15⋊2- 1+4 D12.29D10 D4.10D30
37 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 5A | 5B | 10A | 10B | 10C | ··· | 10H | 20A | 20B | 20C | 20D | 20E | ··· | 20J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
37 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D5 | D10 | D10 | D10 | 2- 1+4 | D4.10D10 |
| kernel | D4.10D10 | C2×Dic10 | C4○D20 | D4⋊2D5 | Q8×D5 | C5×C4○D4 | C4○D4 | C2×C4 | D4 | Q8 | C5 | C1 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 2 | 6 | 6 | 2 | 1 | 4 |
Matrix representation of D4.10D10 ►in GL4(𝔽41) generated by
| 2 | 0 | 6 | 0 |
| 0 | 2 | 0 | 6 |
| 6 | 0 | 39 | 0 |
| 0 | 6 | 0 | 39 |
| 25 | 13 | 19 | 23 |
| 28 | 16 | 18 | 22 |
| 19 | 23 | 16 | 28 |
| 18 | 22 | 13 | 25 |
| 0 | 0 | 7 | 7 |
| 0 | 0 | 34 | 40 |
| 34 | 34 | 0 | 0 |
| 7 | 1 | 0 | 0 |
| 0 | 0 | 14 | 27 |
| 0 | 0 | 11 | 27 |
| 27 | 14 | 0 | 0 |
| 30 | 14 | 0 | 0 |
G:=sub<GL(4,GF(41))| [2,0,6,0,0,2,0,6,6,0,39,0,0,6,0,39],[25,28,19,18,13,16,23,22,19,18,16,13,23,22,28,25],[0,0,34,7,0,0,34,1,7,34,0,0,7,40,0,0],[0,0,27,30,0,0,14,14,14,11,0,0,27,27,0,0] >;
D4.10D10 in GAP, Magma, Sage, TeX
D_4._{10}D_{10} % in TeX
G:=Group("D4.10D10"); // GroupNames label
G:=SmallGroup(160,225);
// by ID
G=gap.SmallGroup(160,225);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,188,86,579,69,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^10=d^2=a^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^9>;
// generators/relations