direct product, metabelian, supersoluble, monomial, A-group
Aliases: Dic5×C2×C10, C102⋊12C4, C102.30C22, (C2×C10)⋊5C20, C10⋊3(C2×C20), C5⋊3(C22×C20), C23.2(C5×D5), (C2×C10).48D10, (C2×C102).4C2, C52⋊12(C22×C4), (C22×C10).8D5, (C5×C10).27C23, C10.9(C22×C10), (C22×C10).5C10, C10.48(C22×D5), C22.11(D5×C10), C2.2(D5×C2×C10), (C5×C10)⋊11(C2×C4), (C2×C10).14(C2×C10), SmallGroup(400,189)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — Dic5×C2×C10 |
Generators and relations for Dic5×C2×C10
G = < a,b,c,d | a2=b10=c10=1, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 260 in 140 conjugacy classes, 86 normal (14 characteristic)
C1, C2, C2, C4, C22, C5, C5, C2×C4, C23, C10, C10, C10, C22×C4, Dic5, C20, C2×C10, C2×C10, C52, C2×Dic5, C2×C20, C22×C10, C22×C10, C5×C10, C5×C10, C22×Dic5, C22×C20, C5×Dic5, C102, C10×Dic5, C2×C102, Dic5×C2×C10
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, D5, C10, C22×C4, Dic5, C20, D10, C2×C10, C2×Dic5, C2×C20, C22×D5, C22×C10, C5×D5, C22×Dic5, C22×C20, C5×Dic5, D5×C10, C10×Dic5, D5×C2×C10, Dic5×C2×C10
►On 80 points
Generators in S80
(1 36)(2 37)(3 38)(4 39)(5 40)(6 31)(7 32)(8 33)(9 34)(10 35)(11 59)(12 60)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 41)(28 42)(29 43)(30 44)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 71)(68 72)(69 73)(70 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 22 9 30 7 28 5 26 3 24)(2 23 10 21 8 29 6 27 4 25)(11 75 13 77 15 79 17 71 19 73)(12 76 14 78 16 80 18 72 20 74)(31 41 39 49 37 47 35 45 33 43)(32 42 40 50 38 48 36 46 34 44)(51 63 53 65 55 67 57 69 59 61)(52 64 54 66 56 68 58 70 60 62)
(1 76 28 18)(2 77 29 19)(3 78 30 20)(4 79 21 11)(5 80 22 12)(6 71 23 13)(7 72 24 14)(8 73 25 15)(9 74 26 16)(10 75 27 17)(31 67 47 51)(32 68 48 52)(33 69 49 53)(34 70 50 54)(35 61 41 55)(36 62 42 56)(37 63 43 57)(38 64 44 58)(39 65 45 59)(40 66 46 60)
G:=sub<Sym(80)| (1,36)(2,37)(3,38)(4,39)(5,40)(6,31)(7,32)(8,33)(9,34)(10,35)(11,59)(12,60)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,41)(28,42)(29,43)(30,44)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,71)(68,72)(69,73)(70,74), (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), (1,22,9,30,7,28,5,26,3,24)(2,23,10,21,8,29,6,27,4,25)(11,75,13,77,15,79,17,71,19,73)(12,76,14,78,16,80,18,72,20,74)(31,41,39,49,37,47,35,45,33,43)(32,42,40,50,38,48,36,46,34,44)(51,63,53,65,55,67,57,69,59,61)(52,64,54,66,56,68,58,70,60,62), (1,76,28,18)(2,77,29,19)(3,78,30,20)(4,79,21,11)(5,80,22,12)(6,71,23,13)(7,72,24,14)(8,73,25,15)(9,74,26,16)(10,75,27,17)(31,67,47,51)(32,68,48,52)(33,69,49,53)(34,70,50,54)(35,61,41,55)(36,62,42,56)(37,63,43,57)(38,64,44,58)(39,65,45,59)(40,66,46,60)>;
G:=Group( (1,36)(2,37)(3,38)(4,39)(5,40)(6,31)(7,32)(8,33)(9,34)(10,35)(11,59)(12,60)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,41)(28,42)(29,43)(30,44)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,71)(68,72)(69,73)(70,74), (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), (1,22,9,30,7,28,5,26,3,24)(2,23,10,21,8,29,6,27,4,25)(11,75,13,77,15,79,17,71,19,73)(12,76,14,78,16,80,18,72,20,74)(31,41,39,49,37,47,35,45,33,43)(32,42,40,50,38,48,36,46,34,44)(51,63,53,65,55,67,57,69,59,61)(52,64,54,66,56,68,58,70,60,62), (1,76,28,18)(2,77,29,19)(3,78,30,20)(4,79,21,11)(5,80,22,12)(6,71,23,13)(7,72,24,14)(8,73,25,15)(9,74,26,16)(10,75,27,17)(31,67,47,51)(32,68,48,52)(33,69,49,53)(34,70,50,54)(35,61,41,55)(36,62,42,56)(37,63,43,57)(38,64,44,58)(39,65,45,59)(40,66,46,60) );
G=PermutationGroup([[(1,36),(2,37),(3,38),(4,39),(5,40),(6,31),(7,32),(8,33),(9,34),(10,35),(11,59),(12,60),(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,57),(20,58),(21,45),(22,46),(23,47),(24,48),(25,49),(26,50),(27,41),(28,42),(29,43),(30,44),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,71),(68,72),(69,73),(70,74)], [(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)], [(1,22,9,30,7,28,5,26,3,24),(2,23,10,21,8,29,6,27,4,25),(11,75,13,77,15,79,17,71,19,73),(12,76,14,78,16,80,18,72,20,74),(31,41,39,49,37,47,35,45,33,43),(32,42,40,50,38,48,36,46,34,44),(51,63,53,65,55,67,57,69,59,61),(52,64,54,66,56,68,58,70,60,62)], [(1,76,28,18),(2,77,29,19),(3,78,30,20),(4,79,21,11),(5,80,22,12),(6,71,23,13),(7,72,24,14),(8,73,25,15),(9,74,26,16),(10,75,27,17),(31,67,47,51),(32,68,48,52),(33,69,49,53),(34,70,50,54),(35,61,41,55),(36,62,42,56),(37,63,43,57),(38,64,44,58),(39,65,45,59),(40,66,46,60)]])
160 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 10A | ··· | 10AB | 10AC | ··· | 10CT | 20A | ··· | 20AF |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 5 | ··· | 5 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 |
160 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C20 | D5 | Dic5 | D10 | C5×D5 | C5×Dic5 | D5×C10 |
| kernel | Dic5×C2×C10 | C10×Dic5 | C2×C102 | C102 | C22×Dic5 | C2×Dic5 | C22×C10 | C2×C10 | C22×C10 | C2×C10 | C2×C10 | C23 | C22 | C22 |
| # reps | 1 | 6 | 1 | 8 | 4 | 24 | 4 | 32 | 2 | 8 | 6 | 8 | 32 | 24 |
Matrix representation of Dic5×C2×C10 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 31 | 0 | 0 | 0 |
| 0 | 31 | 0 | 0 |
| 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 37 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 38 | 37 |
| 9 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 32 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[31,0,0,0,0,31,0,0,0,0,37,0,0,0,0,37],[40,0,0,0,0,1,0,0,0,0,10,38,0,0,0,37],[9,0,0,0,0,40,0,0,0,0,1,0,0,0,32,40] >;
Dic5×C2×C10 in GAP, Magma, Sage, TeX
{\rm Dic}_5\times C_2\times C_{10} % in TeX
G:=Group("Dic5xC2xC10"); // GroupNames label
G:=SmallGroup(400,189);
// by ID
G=gap.SmallGroup(400,189);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-5,240,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^10=c^10=1,d^2=c^5,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations