direct product, abelian, monomial, 2-elementary
Aliases: C22×C20, SmallGroup(80,45)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22×C20 |
| C1 — C22×C20 |
| C1 — C22×C20 |
Generators and relations for C22×C20
G = < a,b,c | a2=b2=c20=1, ab=ba, ac=ca, bc=cb >
Subgroups: 54, all normal (8 characteristic)
C1, C2, C2, C4, C22, C5, C2×C4, C23, C10, C10, C22×C4, C20, C2×C10, C2×C20, C22×C10, C22×C20
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, C10, C22×C4, C20, C2×C10, C2×C20, C22×C10, C22×C20
►Regular action on 80 points
Generators in S80
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 21)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(41 70)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 78)(50 79)(51 80)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 41)(20 42)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 72)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 61)(38 62)(39 63)(40 64)
(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)
G:=sub<Sym(80)| (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,21)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)(51,80)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,41)(20,42)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,61)(38,62)(39,63)(40,64), (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)>;
G:=Group( (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,21)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)(51,80)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,41)(20,42)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,61)(38,62)(39,63)(40,64), (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) );
G=PermutationGroup([[(1,28),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,37),(11,38),(12,39),(13,40),(14,21),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(41,70),(42,71),(43,72),(44,73),(45,74),(46,75),(47,76),(48,77),(49,78),(50,79),(51,80),(52,61),(53,62),(54,63),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(15,57),(16,58),(17,59),(18,60),(19,41),(20,42),(21,65),(22,66),(23,67),(24,68),(25,69),(26,70),(27,71),(28,72),(29,73),(30,74),(31,75),(32,76),(33,77),(34,78),(35,79),(36,80),(37,61),(38,62),(39,63),(40,64)], [(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)]])
C22×C20 is a maximal subgroup of
C20.55D4 C10.10C42 C20.48D4 C23.21D10 C23.23D10 C20⋊7D4
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 20A | ··· | 20AF |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C20 |
| kernel | C22×C20 | C2×C20 | C22×C10 | C2×C10 | C22×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 6 | 1 | 8 | 4 | 24 | 4 | 32 |
Matrix representation of C22×C20 ►in GL3(𝔽41) generated by
| 1 | 0 | 0 |
| 0 | 40 | 0 |
| 0 | 0 | 40 |
| 40 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 40 |
| 16 | 0 | 0 |
| 0 | 25 | 0 |
| 0 | 0 | 5 |
G:=sub<GL(3,GF(41))| [1,0,0,0,40,0,0,0,40],[40,0,0,0,1,0,0,0,40],[16,0,0,0,25,0,0,0,5] >;
C22×C20 in GAP, Magma, Sage, TeX
C_2^2\times C_{20} % in TeX
G:=Group("C2^2xC20"); // GroupNames label
G:=SmallGroup(80,45);
// by ID
G=gap.SmallGroup(80,45);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-2,200]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^20=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations