direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D20⋊4C4, C42⋊37D10, C10⋊3C4≀C2, C4○D20⋊7C4, (C2×C42)⋊5D5, D20⋊26(C2×C4), (C2×D20)⋊17C4, (C2×C4).88D20, C4.81(C2×D20), (C4×C20)⋊52C22, (C2×C20).480D4, C20.301(C2×D4), (C2×Dic10)⋊16C4, Dic10⋊24(C2×C4), C20.92(C22⋊C4), C20.168(C22×C4), (C2×C20).793C23, C4○D20.37C22, (C22×C4).417D10, (C22×C10).179D4, C23.73(C5⋊D4), C4.Dic5⋊19C22, C4.21(D10⋊C4), (C22×C20).536C22, C22.43(D10⋊C4), C5⋊5(C2×C4≀C2), (C2×C4×C20)⋊10C2, C4.67(C2×C4×D5), (C2×C4○D20).3C2, (C2×C4).107(C4×D5), (C2×C4.Dic5)⋊3C2, (C2×C20).395(C2×C4), C2.4(C2×D10⋊C4), (C2×C10).422(C2×D4), C10.71(C2×C22⋊C4), C22.25(C2×C5⋊D4), (C2×C4).236(C5⋊D4), (C2×C4).707(C22×D5), (C2×C10).119(C22⋊C4), SmallGroup(320,554)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D20⋊4C4
G = < a,b,c,d | a2=b20=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b15c >
Subgroups: 606 in 170 conjugacy classes, 63 normal (41 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, C42, C42, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C4≀C2, C2×C42, C2×M4(2), C2×C4○D4, C5⋊2C8, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C2×C4≀C2, C2×C5⋊2C8, C4.Dic5, C4.Dic5, C4×C20, C4×C20, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C4○D20, C2×C5⋊D4, C22×C20, C22×C20, D20⋊4C4, C2×C4.Dic5, C2×C4×C20, C2×C4○D20, C2×D20⋊4C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C4≀C2, C2×C22⋊C4, C4×D5, D20, C5⋊D4, C22×D5, C2×C4≀C2, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, D20⋊4C4, C2×D10⋊C4, C2×D20⋊4C4
►On 80 points
(1 36)(2 37)(3 38)(4 39)(5 40)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(41 68)(42 69)(43 70)(44 71)(45 72)(46 73)(47 74)(48 75)(49 76)(50 77)(51 78)(52 79)(53 80)(54 61)(55 62)(56 63)(57 64)(58 65)(59 66)(60 67)
(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 63)(2 62)(3 61)(4 80)(5 79)(6 78)(7 77)(8 76)(9 75)(10 74)(11 73)(12 72)(13 71)(14 70)(15 69)(16 68)(17 67)(18 66)(19 65)(20 64)(21 51)(22 50)(23 49)(24 48)(25 47)(26 46)(27 45)(28 44)(29 43)(30 42)(31 41)(32 60)(33 59)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 21)(17 22)(18 23)(19 24)(20 25)(41 73 51 63)(42 74 52 64)(43 75 53 65)(44 76 54 66)(45 77 55 67)(46 78 56 68)(47 79 57 69)(48 80 58 70)(49 61 59 71)(50 62 60 72)
G:=sub<Sym(80)| (1,36)(2,37)(3,38)(4,39)(5,40)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,61)(55,62)(56,63)(57,64)(58,65)(59,66)(60,67), (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), (1,63)(2,62)(3,61)(4,80)(5,79)(6,78)(7,77)(8,76)(9,75)(10,74)(11,73)(12,72)(13,71)(14,70)(15,69)(16,68)(17,67)(18,66)(19,65)(20,64)(21,51)(22,50)(23,49)(24,48)(25,47)(26,46)(27,45)(28,44)(29,43)(30,42)(31,41)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52), (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,21)(17,22)(18,23)(19,24)(20,25)(41,73,51,63)(42,74,52,64)(43,75,53,65)(44,76,54,66)(45,77,55,67)(46,78,56,68)(47,79,57,69)(48,80,58,70)(49,61,59,71)(50,62,60,72)>;
G:=Group( (1,36)(2,37)(3,38)(4,39)(5,40)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,61)(55,62)(56,63)(57,64)(58,65)(59,66)(60,67), (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), (1,63)(2,62)(3,61)(4,80)(5,79)(6,78)(7,77)(8,76)(9,75)(10,74)(11,73)(12,72)(13,71)(14,70)(15,69)(16,68)(17,67)(18,66)(19,65)(20,64)(21,51)(22,50)(23,49)(24,48)(25,47)(26,46)(27,45)(28,44)(29,43)(30,42)(31,41)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52), (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,21)(17,22)(18,23)(19,24)(20,25)(41,73,51,63)(42,74,52,64)(43,75,53,65)(44,76,54,66)(45,77,55,67)(46,78,56,68)(47,79,57,69)(48,80,58,70)(49,61,59,71)(50,62,60,72) );
G=PermutationGroup([[(1,36),(2,37),(3,38),(4,39),(5,40),(6,21),(7,22),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30),(16,31),(17,32),(18,33),(19,34),(20,35),(41,68),(42,69),(43,70),(44,71),(45,72),(46,73),(47,74),(48,75),(49,76),(50,77),(51,78),(52,79),(53,80),(54,61),(55,62),(56,63),(57,64),(58,65),(59,66),(60,67)], [(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)], [(1,63),(2,62),(3,61),(4,80),(5,79),(6,78),(7,77),(8,76),(9,75),(10,74),(11,73),(12,72),(13,71),(14,70),(15,69),(16,68),(17,67),(18,66),(19,65),(20,64),(21,51),(22,50),(23,49),(24,48),(25,47),(26,46),(27,45),(28,44),(29,43),(30,42),(31,41),(32,60),(33,59),(34,58),(35,57),(36,56),(37,55),(38,54),(39,53),(40,52)], [(1,26),(2,27),(3,28),(4,29),(5,30),(6,31),(7,32),(8,33),(9,34),(10,35),(11,36),(12,37),(13,38),(14,39),(15,40),(16,21),(17,22),(18,23),(19,24),(20,25),(41,73,51,63),(42,74,52,64),(43,75,53,65),(44,76,54,66),(45,77,55,67),(46,78,56,68),(47,79,57,69),(48,80,58,70),(49,61,59,71),(50,62,60,72)]])
92 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10N | 20A | ··· | 20AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | D5 | D10 | D10 | C4≀C2 | C4×D5 | D20 | C5⋊D4 | C5⋊D4 | D20⋊4C4 |
| kernel | C2×D20⋊4C4 | D20⋊4C4 | C2×C4.Dic5 | C2×C4×C20 | C2×C4○D20 | C2×Dic10 | C2×D20 | C4○D20 | C2×C20 | C22×C10 | C2×C42 | C42 | C22×C4 | C10 | C2×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 3 | 1 | 2 | 4 | 2 | 8 | 8 | 8 | 4 | 4 | 32 |
Matrix representation of C2×D20⋊4C4 ►in GL3(𝔽41) generated by
| 40 | 0 | 0 |
| 0 | 40 | 0 |
| 0 | 0 | 40 |
| 1 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 33 |
| 1 | 0 | 0 |
| 0 | 0 | 8 |
| 0 | 36 | 0 |
| 9 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 9 |
G:=sub<GL(3,GF(41))| [40,0,0,0,40,0,0,0,40],[1,0,0,0,5,0,0,0,33],[1,0,0,0,0,36,0,8,0],[9,0,0,0,1,0,0,0,9] >;
C2×D20⋊4C4 in GAP, Magma, Sage, TeX
C_2\times D_{20}\rtimes_4C_4 % in TeX
G:=Group("C2xD20:4C4"); // GroupNames label
G:=SmallGroup(320,554);
// by ID
G=gap.SmallGroup(320,554);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,422,58,1123,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^15*c>;
// generators/relations