metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.3C4, C8.18D10, C40.23C22, C20.37C23, Dic10.3C4, (C2×C8)⋊7D5, (C8×D5)⋊6C2, C5⋊3(C8○D4), (C2×C40)⋊10C2, C8⋊D5⋊7C2, C4.10(C4×D5), C5⋊D4.3C4, C20.41(C2×C4), C4○D20.6C2, D10.3(C2×C4), (C2×C4).78D10, C22.2(C4×D5), C4.Dic5⋊11C2, Dic5.5(C2×C4), C4.37(C22×D5), (C2×C20).98C22, C10.27(C22×C4), C5⋊2C8.11C22, (C4×D5).23C22, C2.15(C2×C4×D5), (C2×C10).36(C2×C4), SmallGroup(160,122)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.3C4
G = < a,b,c | a20=b2=1, c4=a10, bab=a-1, ac=ca, bc=cb >
Subgroups: 160 in 62 conjugacy classes, 37 normal (23 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, C2×C8, C2×C8, 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, C4.Dic5, C2×C40, C4○D20, D20.3C4
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, D10, C8○D4, C4×D5, C22×D5, C2×C4×D5, D20.3C4
►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 25)(22 24)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(41 47)(42 46)(43 45)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(61 67)(62 66)(63 65)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)
(1 67 57 26 11 77 47 36)(2 68 58 27 12 78 48 37)(3 69 59 28 13 79 49 38)(4 70 60 29 14 80 50 39)(5 71 41 30 15 61 51 40)(6 72 42 31 16 62 52 21)(7 73 43 32 17 63 53 22)(8 74 44 33 18 64 54 23)(9 75 45 34 19 65 55 24)(10 76 46 35 20 66 56 25)
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)(21,25)(22,24)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(41,47)(42,46)(43,45)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(61,67)(62,66)(63,65)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75), (1,67,57,26,11,77,47,36)(2,68,58,27,12,78,48,37)(3,69,59,28,13,79,49,38)(4,70,60,29,14,80,50,39)(5,71,41,30,15,61,51,40)(6,72,42,31,16,62,52,21)(7,73,43,32,17,63,53,22)(8,74,44,33,18,64,54,23)(9,75,45,34,19,65,55,24)(10,76,46,35,20,66,56,25)>;
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)(21,25)(22,24)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(41,47)(42,46)(43,45)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(61,67)(62,66)(63,65)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75), (1,67,57,26,11,77,47,36)(2,68,58,27,12,78,48,37)(3,69,59,28,13,79,49,38)(4,70,60,29,14,80,50,39)(5,71,41,30,15,61,51,40)(6,72,42,31,16,62,52,21)(7,73,43,32,17,63,53,22)(8,74,44,33,18,64,54,23)(9,75,45,34,19,65,55,24)(10,76,46,35,20,66,56,25) );
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),(21,25),(22,24),(26,40),(27,39),(28,38),(29,37),(30,36),(31,35),(32,34),(41,47),(42,46),(43,45),(48,60),(49,59),(50,58),(51,57),(52,56),(53,55),(61,67),(62,66),(63,65),(68,80),(69,79),(70,78),(71,77),(72,76),(73,75)], [(1,67,57,26,11,77,47,36),(2,68,58,27,12,78,48,37),(3,69,59,28,13,79,49,38),(4,70,60,29,14,80,50,39),(5,71,41,30,15,61,51,40),(6,72,42,31,16,62,52,21),(7,73,43,32,17,63,53,22),(8,74,44,33,18,64,54,23),(9,75,45,34,19,65,55,24),(10,76,46,35,20,66,56,25)]])
D20.3C4 is a maximal subgroup of
D20.3C8 D20.4C8 D40⋊17C4 D40⋊10C4 C8.20D20 C8.21D20 C8.24D20 D20.6C8 D20.5C8 C40.93D4 C40.50D4 C40.23D4 C40.44D4 C40.29D4 C40.47C23 D5×C8○D4 C20.72C24 D8⋊13D10 D20.29D4 D20.30D4 D8⋊15D10 D8⋊11D10 D20.47D4 C40.54D6 C40.55D6 D20.3Dic3 D60.4C4 D60.6C4
D20.3C4 is a maximal quotient of
C8×Dic10 C40⋊11Q8 C8×D20 C8⋊6D20 D10.5C42 C42.243D10 C40⋊8C4⋊C2 C22⋊C8⋊D5 D10⋊4M4(2) Dic5⋊2M4(2) Dic5.5M4(2) D10⋊5M4(2) C42.30D10 C42.31D10 C20.42C42 C20.65(C4⋊C4) C8×C5⋊D4 (C22×C8)⋊D5 C40⋊32D4 C40.54D6 C40.55D6 D20.3Dic3 D60.4C4 D60.6C4
52 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 | ··· | 10F | 20A | ··· | 20H | 40A | ··· | 40P |
| 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 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 10 | 10 | 1 | 1 | 2 | 10 | 10 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D5 | D10 | D10 | C8○D4 | C4×D5 | C4×D5 | D20.3C4 |
| kernel | D20.3C4 | C8×D5 | C8⋊D5 | C4.Dic5 | C2×C40 | C4○D20 | Dic10 | D20 | C5⋊D4 | C2×C8 | C8 | C2×C4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 2 | 4 | 4 | 4 | 16 |
Matrix representation of D20.3C4 ►in GL2(𝔽41) generated by
| 14 | 39 |
| 16 | 30 |
| 1 | 1 |
| 0 | 40 |
| 14 | 0 |
| 0 | 14 |
G:=sub<GL(2,GF(41))| [14,16,39,30],[1,0,1,40],[14,0,0,14] >;
D20.3C4 in GAP, Magma, Sage, TeX
D_{20}._3C_4 % in TeX
G:=Group("D20.3C4"); // GroupNames label
G:=SmallGroup(160,122);
// by ID
G=gap.SmallGroup(160,122);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,50,69,4613]);
// Polycyclic
G:=Group<a,b,c|a^20=b^2=1,c^4=a^10,b*a*b=a^-1,a*c=c*a,b*c=c*b>;
// generators/relations