metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.2C4, C8.12D10, M4(2)⋊5D5, C40.12C22, C20.39C23, Dic10.2C4, (C8×D5)⋊8C2, C5⋊4(C8○D4), C4.5(C4×D5), C8⋊D5⋊6C2, C5⋊D4.2C4, C20.34(C2×C4), C4○D20.3C2, D10.4(C2×C4), (C2×C4).46D10, C22.1(C4×D5), (C5×M4(2))⋊4C2, Dic5.6(C2×C4), C4.39(C22×D5), (C2×C20).26C22, C10.29(C22×C4), C5⋊2C8.12C22, (C4×D5).24C22, (C2×C5⋊2C8)⋊3C2, C2.17(C2×C4×D5), (C2×C10).26(C2×C4), SmallGroup(160,128)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.2C4
G = < a,b,c | a20=b2=1, c4=a10, bab=a-1, cac-1=a11, cbc-1=a10b >
Subgroups: 160 in 62 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, C2×C8, M4(2), M4(2), C4○D4, Dic5, C20, D10, C2×C10, C8○D4, C5⋊2C8, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C8×D5, C8⋊D5, C2×C5⋊2C8, C5×M4(2), C4○D20, D20.2C4
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, D10, C8○D4, C4×D5, C22×D5, C2×C4×D5, D20.2C4
►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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 23)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(41 51)(42 50)(43 49)(44 48)(45 47)(52 60)(53 59)(54 58)(55 57)(61 73)(62 72)(63 71)(64 70)(65 69)(66 68)(74 80)(75 79)(76 78)
(1 75 59 30 11 65 49 40)(2 66 60 21 12 76 50 31)(3 77 41 32 13 67 51 22)(4 68 42 23 14 78 52 33)(5 79 43 34 15 69 53 24)(6 70 44 25 16 80 54 35)(7 61 45 36 17 71 55 26)(8 72 46 27 18 62 56 37)(9 63 47 38 19 73 57 28)(10 74 48 29 20 64 58 39)
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,23)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(41,51)(42,50)(43,49)(44,48)(45,47)(52,60)(53,59)(54,58)(55,57)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(74,80)(75,79)(76,78), (1,75,59,30,11,65,49,40)(2,66,60,21,12,76,50,31)(3,77,41,32,13,67,51,22)(4,68,42,23,14,78,52,33)(5,79,43,34,15,69,53,24)(6,70,44,25,16,80,54,35)(7,61,45,36,17,71,55,26)(8,72,46,27,18,62,56,37)(9,63,47,38,19,73,57,28)(10,74,48,29,20,64,58,39)>;
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,23)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(41,51)(42,50)(43,49)(44,48)(45,47)(52,60)(53,59)(54,58)(55,57)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(74,80)(75,79)(76,78), (1,75,59,30,11,65,49,40)(2,66,60,21,12,76,50,31)(3,77,41,32,13,67,51,22)(4,68,42,23,14,78,52,33)(5,79,43,34,15,69,53,24)(6,70,44,25,16,80,54,35)(7,61,45,36,17,71,55,26)(8,72,46,27,18,62,56,37)(9,63,47,38,19,73,57,28)(10,74,48,29,20,64,58,39) );
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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,23),(24,40),(25,39),(26,38),(27,37),(28,36),(29,35),(30,34),(31,33),(41,51),(42,50),(43,49),(44,48),(45,47),(52,60),(53,59),(54,58),(55,57),(61,73),(62,72),(63,71),(64,70),(65,69),(66,68),(74,80),(75,79),(76,78)], [(1,75,59,30,11,65,49,40),(2,66,60,21,12,76,50,31),(3,77,41,32,13,67,51,22),(4,68,42,23,14,78,52,33),(5,79,43,34,15,69,53,24),(6,70,44,25,16,80,54,35),(7,61,45,36,17,71,55,26),(8,72,46,27,18,62,56,37),(9,63,47,38,19,73,57,28),(10,74,48,29,20,64,58,39)]])
D20.2C4 is a maximal subgroup of
D20.C8 D20.2D4 D20.3D4 D20.6D4 D20.7D4 M4(2).22D10 C42.196D10 D40⋊16C4 D40⋊13C4 Dic10.C8 C40.47C23 D5×C8○D4 C20.72C24 D8⋊5D10 D8⋊6D10 C40.C23 D20.44D4 C40.34D6 C40.35D6 D20.2Dic3 D60.5C4 D60.3C4
D20.2C4 is a maximal quotient of
C40⋊Q8 C8⋊9D20 D10.7C42 C42.185D10 C40⋊8C4⋊C2 C5⋊5(C8×D4) C22⋊C8⋊D5 C5⋊2C8⋊26D4 Dic10⋊5C8 C42.198D10 D20⋊5C8 C20⋊6M4(2) C42.30D10 C42.31D10 C20.51(C4⋊C4) C20.37C42 C40⋊D4 C40⋊18D4 C4.89(C2×D20) C40.34D6 C40.35D6 D20.2Dic3 D60.5C4 D60.3C4
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10A | 10B | 10C | 10D | 20A | 20B | 20C | 20D | 20E | 20F | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 10 | 10 | 1 | 1 | 2 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D5 | D10 | D10 | C8○D4 | C4×D5 | C4×D5 | D20.2C4 |
| kernel | D20.2C4 | C8×D5 | C8⋊D5 | C2×C5⋊2C8 | C5×M4(2) | C4○D20 | Dic10 | D20 | C5⋊D4 | M4(2) | C8 | C2×C4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 2 | 4 | 4 | 4 | 4 |
Matrix representation of D20.2C4 ►in GL4(𝔽41) generated by
| 0 | 40 | 0 | 0 |
| 1 | 35 | 0 | 0 |
| 0 | 0 | 27 | 34 |
| 0 | 0 | 34 | 14 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 4 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 21 |
| 0 | 0 | 16 | 40 |
G:=sub<GL(4,GF(41))| [0,1,0,0,40,35,0,0,0,0,27,34,0,0,34,14],[0,1,0,0,1,0,0,0,0,0,40,4,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,16,0,0,21,40] >;
D20.2C4 in GAP, Magma, Sage, TeX
D_{20}._2C_4 % in TeX
G:=Group("D20.2C4"); // GroupNames label
G:=SmallGroup(160,128);
// by ID
G=gap.SmallGroup(160,128);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,188,50,69,4613]);
// Polycyclic
G:=Group<a,b,c|a^20=b^2=1,c^4=a^10,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^10*b>;
// generators/relations