direct product, abelian, monomial, 2-elementary
Aliases: C22×C10, SmallGroup(40,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22×C10 |
| C1 — C22×C10 |
| C1 — C22×C10 |
Generators and relations for C22×C10
G = < a,b,c | a2=b2=c10=1, ab=ba, ac=ca, bc=cb >
Smallest permutation representation of C22×C10
►Regular action on 40 points
Generators in S40
(1 27)(2 28)(3 29)(4 30)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)
(1 17)(2 18)(3 19)(4 20)(5 11)(6 12)(7 13)(8 14)(9 15)(10 16)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(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)
G:=sub<Sym(40)| (1,27)(2,28)(3,29)(4,30)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40), (1,17)(2,18)(3,19)(4,20)(5,11)(6,12)(7,13)(8,14)(9,15)(10,16)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (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)>;
G:=Group( (1,27)(2,28)(3,29)(4,30)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40), (1,17)(2,18)(3,19)(4,20)(5,11)(6,12)(7,13)(8,14)(9,15)(10,16)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (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) );
G=PermutationGroup([[(1,27),(2,28),(3,29),(4,30),(5,21),(6,22),(7,23),(8,24),(9,25),(10,26),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40)], [(1,17),(2,18),(3,19),(4,20),(5,11),(6,12),(7,13),(8,14),(9,15),(10,16),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40)], [(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)]])
C22×C10 is a maximal subgroup of
C23.D5
40 conjugacy classes
| class | 1 | 2A | ··· | 2G | 5A | 5B | 5C | 5D | 10A | ··· | 10AB |
| order | 1 | 2 | ··· | 2 | 5 | 5 | 5 | 5 | 10 | ··· | 10 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C5 | C10 |
| kernel | C22×C10 | C2×C10 | C23 | C22 |
| # reps | 1 | 7 | 4 | 28 |
Matrix representation of C22×C10 ►in GL3(𝔽11) generated by
| 10 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 10 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 2 |
G:=sub<GL(3,GF(11))| [10,0,0,0,1,0,0,0,1],[10,0,0,0,10,0,0,0,1],[1,0,0,0,1,0,0,0,2] >;
C22×C10 in GAP, Magma, Sage, TeX
C_2^2\times C_{10} % in TeX
G:=Group("C2^2xC10"); // GroupNames label
G:=SmallGroup(40,14);
// by ID
G=gap.SmallGroup(40,14);
# by ID
G:=PCGroup([4,-2,-2,-2,-5]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^10=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations
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