direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4○D20, C10.4C24, D20⋊12C22, C20.43C23, D10.1C23, C23.26D10, Dic5.2C23, Dic10⋊11C22, (C2×C4)⋊10D10, (C22×C4)⋊6D5, (C2×D20)⋊14C2, C10⋊1(C4○D4), (C22×C20)⋊8C2, (C4×D5)⋊6C22, C5⋊D4⋊6C22, C2.5(C23×D5), (C2×C20)⋊13C22, C4.43(C22×D5), (C2×Dic10)⋊15C2, (C2×C10).65C23, C22.5(C22×D5), (C22×C10).46C22, (C2×Dic5).46C22, (C22×D5).31C22, C5⋊1(C2×C4○D4), (C2×C4×D5)⋊15C2, (C2×C5⋊D4)⋊12C2, SmallGroup(160,216)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C4○D20
G = < a,b,c,d | a2=b4=d2=1, c10=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c9 >
Subgroups: 456 in 164 conjugacy classes, 89 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C22×C20, C2×C4○D20
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, C22×D5, C4○D20, C23×D5, C2×C4○D20
►On 80 points
(1 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 40)(18 21)(19 22)(20 23)(41 79)(42 80)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)
(1 79 11 69)(2 80 12 70)(3 61 13 71)(4 62 14 72)(5 63 15 73)(6 64 16 74)(7 65 17 75)(8 66 18 76)(9 67 19 77)(10 68 20 78)(21 58 31 48)(22 59 32 49)(23 60 33 50)(24 41 34 51)(25 42 35 52)(26 43 36 53)(27 44 37 54)(28 45 38 55)(29 46 39 56)(30 47 40 57)
(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 23)(2 22)(3 21)(4 40)(5 39)(6 38)(7 37)(8 36)(9 35)(10 34)(11 33)(12 32)(13 31)(14 30)(15 29)(16 28)(17 27)(18 26)(19 25)(20 24)(41 78)(42 77)(43 76)(44 75)(45 74)(46 73)(47 72)(48 71)(49 70)(50 69)(51 68)(52 67)(53 66)(54 65)(55 64)(56 63)(57 62)(58 61)(59 80)(60 79)
G:=sub<Sym(80)| (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,21)(19,22)(20,23)(41,79)(42,80)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78), (1,79,11,69)(2,80,12,70)(3,61,13,71)(4,62,14,72)(5,63,15,73)(6,64,16,74)(7,65,17,75)(8,66,18,76)(9,67,19,77)(10,68,20,78)(21,58,31,48)(22,59,32,49)(23,60,33,50)(24,41,34,51)(25,42,35,52)(26,43,36,53)(27,44,37,54)(28,45,38,55)(29,46,39,56)(30,47,40,57), (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), (1,23)(2,22)(3,21)(4,40)(5,39)(6,38)(7,37)(8,36)(9,35)(10,34)(11,33)(12,32)(13,31)(14,30)(15,29)(16,28)(17,27)(18,26)(19,25)(20,24)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,72)(48,71)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,80)(60,79)>;
G:=Group( (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,21)(19,22)(20,23)(41,79)(42,80)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78), (1,79,11,69)(2,80,12,70)(3,61,13,71)(4,62,14,72)(5,63,15,73)(6,64,16,74)(7,65,17,75)(8,66,18,76)(9,67,19,77)(10,68,20,78)(21,58,31,48)(22,59,32,49)(23,60,33,50)(24,41,34,51)(25,42,35,52)(26,43,36,53)(27,44,37,54)(28,45,38,55)(29,46,39,56)(30,47,40,57), (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), (1,23)(2,22)(3,21)(4,40)(5,39)(6,38)(7,37)(8,36)(9,35)(10,34)(11,33)(12,32)(13,31)(14,30)(15,29)(16,28)(17,27)(18,26)(19,25)(20,24)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,72)(48,71)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,80)(60,79) );
G=PermutationGroup([[(1,24),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,31),(9,32),(10,33),(11,34),(12,35),(13,36),(14,37),(15,38),(16,39),(17,40),(18,21),(19,22),(20,23),(41,79),(42,80),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72),(55,73),(56,74),(57,75),(58,76),(59,77),(60,78)], [(1,79,11,69),(2,80,12,70),(3,61,13,71),(4,62,14,72),(5,63,15,73),(6,64,16,74),(7,65,17,75),(8,66,18,76),(9,67,19,77),(10,68,20,78),(21,58,31,48),(22,59,32,49),(23,60,33,50),(24,41,34,51),(25,42,35,52),(26,43,36,53),(27,44,37,54),(28,45,38,55),(29,46,39,56),(30,47,40,57)], [(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)], [(1,23),(2,22),(3,21),(4,40),(5,39),(6,38),(7,37),(8,36),(9,35),(10,34),(11,33),(12,32),(13,31),(14,30),(15,29),(16,28),(17,27),(18,26),(19,25),(20,24),(41,78),(42,77),(43,76),(44,75),(45,74),(46,73),(47,72),(48,71),(49,70),(50,69),(51,68),(52,67),(53,66),(54,65),(55,64),(56,63),(57,62),(58,61),(59,80),(60,79)]])
C2×C4○D20 is a maximal subgroup of
C22⋊C8⋊D5 D20.32D4 D20⋊14D4 C4○D20⋊9C4 C4○D20⋊10C4 C42⋊4D10 (C2×D20)⋊25C4 D20⋊17D4 D20.37D4 (C22×C8)⋊D5 C23.23D20 C4.89(C2×D20) M4(2).31D10 C23.49D20 C23.20D20 C23⋊F5⋊5C2 (C4×D5).D4 C42.276D10 C24.27D10 C10.82+ 1+4 C10.2- 1+4 C10.2+ 1+4 C42.188D10 C42.91D10 C42⋊8D10 C42⋊9D10 C42.92D10 C42⋊12D10 C42.228D10 D20⋊23D4 D20⋊24D4 Dic10⋊23D4 Dic10⋊24D4 Dic10⋊20D4 C10.382+ 1+4 C10.392+ 1+4 D20⋊20D4 C10.162- 1+4 C10.172- 1+4 D20⋊22D4 Dic10⋊22D4 C10.1212+ 1+4 C10.822- 1+4 C40.47C23 C40.9C23 C24.72D10 C24.41D10 C10.442- 1+4 C20.C24 (C2×C20)⋊17D4 C10.1472+ 1+4 C2×D5×C4○D4 C10.C25
C2×C4○D20 is a maximal quotient of
C2×C4×Dic10 C42.274D10 C2×C4×D20 C42.276D10 C42.277D10 C24.27D10 C24.30D10 C24.31D10 C10.2- 1+4 C10.102+ 1+4 C10.52- 1+4 C10.112+ 1+4 C10.62- 1+4 C42.89D10 C42⋊10D10 C42.93D10 C42.94D10 C42.95D10 C42.96D10 C42.97D10 C42.98D10 C42.99D10 C42.100D10 C42.102D10 C42.104D10 C42.105D10 C42.106D10 C42⋊12D10 C42.228D10 D20⋊23D4 D20⋊24D4 Dic10⋊23D4 Dic10⋊24D4 C42⋊16D10 C42.229D10 C42.113D10 C42.114D10 C42⋊17D10 C42.115D10 C42.116D10 C42.117D10 C42.118D10 C42.119D10 Dic10⋊10Q8 C42.122D10 C42.232D10 D20⋊10Q8 C42.131D10 C42.132D10 C42.133D10 C42.134D10 C42.135D10 C42.136D10 C2×C4×C5⋊D4 C24.72D10
52 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 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 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | C4○D20 |
| kernel | C2×C4○D20 | C2×Dic10 | C2×C4×D5 | C2×D20 | C4○D20 | C2×C5⋊D4 | C22×C20 | C22×C4 | C10 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 2 | 1 | 8 | 2 | 1 | 2 | 4 | 12 | 2 | 16 |
Matrix representation of C2×C4○D20 ►in GL3(𝔽41) generated by
| 40 | 0 | 0 |
| 0 | 40 | 0 |
| 0 | 0 | 40 |
| 1 | 0 | 0 |
| 0 | 32 | 0 |
| 0 | 0 | 32 |
| 40 | 0 | 0 |
| 0 | 39 | 27 |
| 0 | 30 | 25 |
| 40 | 0 | 0 |
| 0 | 39 | 4 |
| 0 | 30 | 2 |
G:=sub<GL(3,GF(41))| [40,0,0,0,40,0,0,0,40],[1,0,0,0,32,0,0,0,32],[40,0,0,0,39,30,0,27,25],[40,0,0,0,39,30,0,4,2] >;
C2×C4○D20 in GAP, Magma, Sage, TeX
C_2\times C_4\circ D_{20} % in TeX
G:=Group("C2xC4oD20"); // GroupNames label
G:=SmallGroup(160,216);
// by ID
G=gap.SmallGroup(160,216);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,86,579,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=d^2=1,c^10=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^9>;
// generators/relations