direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×D20, C20⋊5D4, C42⋊4D5, C5⋊2(C4×D4), C4⋊2(C4×D5), C20⋊7(C2×C4), (C4×C20)⋊7C2, D10⋊2(C2×C4), C2.1(C2×D20), C10.2(C2×D4), C4⋊Dic5⋊16C2, (C2×C4).75D10, (C2×D20).11C2, C10.4(C4○D4), C2.3(C4○D20), D10⋊C4⋊17C2, (C2×C20).86C22, C10.17(C22×C4), (C2×C10).14C23, C22.11(C22×D5), (C2×Dic5).28C22, (C22×D5).18C22, (C2×C4×D5)⋊7C2, C2.6(C2×C4×D5), SmallGroup(160,94)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×D20
G = < a,b,c | a4=b20=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 312 in 94 conjugacy classes, 45 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, D4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, C20, D10, D10, C2×C10, C4×D4, C4×D5, D20, C2×Dic5, C2×C20, C22×D5, C4⋊Dic5, D10⋊C4, C4×C20, C2×C4×D5, C2×D20, C4×D20
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, D10, C4×D4, C4×D5, D20, C22×D5, C2×C4×D5, C2×D20, C4○D20, C4×D20
►On 80 points
(1 31 47 64)(2 32 48 65)(3 33 49 66)(4 34 50 67)(5 35 51 68)(6 36 52 69)(7 37 53 70)(8 38 54 71)(9 39 55 72)(10 40 56 73)(11 21 57 74)(12 22 58 75)(13 23 59 76)(14 24 60 77)(15 25 41 78)(16 26 42 79)(17 27 43 80)(18 28 44 61)(19 29 45 62)(20 30 46 63)
(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 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(36 40)(37 39)(41 47)(42 46)(43 45)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)
G:=sub<Sym(80)| (1,31,47,64)(2,32,48,65)(3,33,49,66)(4,34,50,67)(5,35,51,68)(6,36,52,69)(7,37,53,70)(8,38,54,71)(9,39,55,72)(10,40,56,73)(11,21,57,74)(12,22,58,75)(13,23,59,76)(14,24,60,77)(15,25,41,78)(16,26,42,79)(17,27,43,80)(18,28,44,61)(19,29,45,62)(20,30,46,63), (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)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(36,40)(37,39)(41,47)(42,46)(43,45)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)>;
G:=Group( (1,31,47,64)(2,32,48,65)(3,33,49,66)(4,34,50,67)(5,35,51,68)(6,36,52,69)(7,37,53,70)(8,38,54,71)(9,39,55,72)(10,40,56,73)(11,21,57,74)(12,22,58,75)(13,23,59,76)(14,24,60,77)(15,25,41,78)(16,26,42,79)(17,27,43,80)(18,28,44,61)(19,29,45,62)(20,30,46,63), (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)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(36,40)(37,39)(41,47)(42,46)(43,45)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72) );
G=PermutationGroup([[(1,31,47,64),(2,32,48,65),(3,33,49,66),(4,34,50,67),(5,35,51,68),(6,36,52,69),(7,37,53,70),(8,38,54,71),(9,39,55,72),(10,40,56,73),(11,21,57,74),(12,22,58,75),(13,23,59,76),(14,24,60,77),(15,25,41,78),(16,26,42,79),(17,27,43,80),(18,28,44,61),(19,29,45,62),(20,30,46,63)], [(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),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(36,40),(37,39),(41,47),(42,46),(43,45),(48,60),(49,59),(50,58),(51,57),(52,56),(53,55),(62,80),(63,79),(64,78),(65,77),(66,76),(67,75),(68,74),(69,73),(70,72)]])
C4×D20 is a maximal subgroup of
D20⋊3C8 D20⋊4C8 C8⋊6D20 C8⋊9D20 C42.16D10 D40⋊9C4 D20⋊5C8 D10⋊5M4(2) C20⋊6M4(2) C20⋊SD16 D20⋊3Q8 C4⋊D40 D20.19D4 D20⋊4Q8 D20.3Q8 C42.48D10 C42.56D10 D20.23D4 D20.4Q8 C20⋊2D8 C20⋊5SD16 D20⋊5Q8 D20⋊6Q8 C42.276D10 C42.277D10 C42⋊7D10 C42.91D10 C42⋊8D10 C42⋊10D10 C42.93D10 C42.95D10 C42.99D10 C42.100D10 C4×D4×D5 C42⋊11D10 C42⋊12D10 C42.228D10 D20⋊23D4 D20⋊24D4 D4⋊5D20 D4⋊6D20 C42.113D10 C42.116D10 C42.117D10 C42.119D10 C42.126D10 Q8⋊5D20 Q8⋊6D20 D20⋊10Q8 C42.131D10 C42.132D10 C42.133D10 C42.135D10 C42.136D10 D20⋊10D4 Dic10⋊10D4 C42⋊20D10 C42.143D10 D20⋊7Q8 C42.150D10 C42.152D10 C42.153D10 C42⋊23D10 C42⋊24D10 C42.161D10 C42.163D10 D20⋊11D4 Dic10⋊11D4 D20⋊12D4 D20⋊8Q8 D20⋊9Q8 C42.177D10 C42.179D10 Dic3⋊4D20 D60⋊14C4
C4×D20 is a maximal quotient of
C2.(C4×D20) (C2×C4)⋊9D20 D10⋊3(C4⋊C4) C10.55(C4×D4) C8⋊6D20 D40⋊17C4 C8⋊9D20 C42.16D10 D40⋊9C4 Dic20⋊9C4 D40⋊10C4 C20⋊7(C4⋊C4) (C2×C4)⋊6D20 (C2×C42)⋊D5 Dic3⋊4D20 D60⋊14C4
52 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | C4○D4 | D10 | C4×D5 | D20 | C4○D20 |
| kernel | C4×D20 | C4⋊Dic5 | D10⋊C4 | C4×C20 | C2×C4×D5 | C2×D20 | D20 | C20 | C42 | C10 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 2 | 2 | 2 | 6 | 8 | 8 | 8 |
Matrix representation of C4×D20 ►in GL3(𝔽41) generated by
| 32 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 9 |
| 1 | 0 | 0 |
| 0 | 32 | 30 |
| 0 | 11 | 27 |
| 40 | 0 | 0 |
| 0 | 40 | 0 |
| 0 | 7 | 1 |
G:=sub<GL(3,GF(41))| [32,0,0,0,9,0,0,0,9],[1,0,0,0,32,11,0,30,27],[40,0,0,0,40,7,0,0,1] >;
C4×D20 in GAP, Magma, Sage, TeX
C_4\times D_{20} % in TeX
G:=Group("C4xD20"); // GroupNames label
G:=SmallGroup(160,94);
// by ID
G=gap.SmallGroup(160,94);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,103,50,4613]);
// Polycyclic
G:=Group<a,b,c|a^4=b^20=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations