direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C23×D5, C5⋊C24, C10⋊C23, (C22×C10)⋊3C2, (C2×C10)⋊4C22, SmallGroup(80,51)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C23×D5 |
Generators and relations for C23×D5
G = < a,b,c,d,e | a2=b2=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 338 in 134 conjugacy classes, 83 normal (5 characteristic)
C1, C2, C2, C22, C22, C5, C23, C23, D5, C10, C24, D10, C2×C10, C22×D5, C22×C10, C23×D5
Quotients: C1, C2, C22, C23, D5, C24, D10, C22×D5, C23×D5
►On 40 points
Generators in S40
(1 24)(2 25)(3 21)(4 22)(5 23)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)
(1 19)(2 20)(3 16)(4 17)(5 18)(6 11)(7 12)(8 13)(9 14)(10 15)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 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)
(1 23)(2 22)(3 21)(4 25)(5 24)(6 26)(7 30)(8 29)(9 28)(10 27)(11 31)(12 35)(13 34)(14 33)(15 32)(16 36)(17 40)(18 39)(19 38)(20 37)
G:=sub<Sym(40)| (1,24)(2,25)(3,21)(4,22)(5,23)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40), (1,19)(2,20)(3,16)(4,17)(5,18)(6,11)(7,12)(8,13)(9,14)(10,15)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35), (1,9)(2,10)(3,6)(4,7)(5,8)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (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), (1,23)(2,22)(3,21)(4,25)(5,24)(6,26)(7,30)(8,29)(9,28)(10,27)(11,31)(12,35)(13,34)(14,33)(15,32)(16,36)(17,40)(18,39)(19,38)(20,37)>;
G:=Group( (1,24)(2,25)(3,21)(4,22)(5,23)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40), (1,19)(2,20)(3,16)(4,17)(5,18)(6,11)(7,12)(8,13)(9,14)(10,15)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35), (1,9)(2,10)(3,6)(4,7)(5,8)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (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), (1,23)(2,22)(3,21)(4,25)(5,24)(6,26)(7,30)(8,29)(9,28)(10,27)(11,31)(12,35)(13,34)(14,33)(15,32)(16,36)(17,40)(18,39)(19,38)(20,37) );
G=PermutationGroup([[(1,24),(2,25),(3,21),(4,22),(5,23),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40)], [(1,19),(2,20),(3,16),(4,17),(5,18),(6,11),(7,12),(8,13),(9,14),(10,15),(21,36),(22,37),(23,38),(24,39),(25,40),(26,31),(27,32),(28,33),(29,34),(30,35)], [(1,9),(2,10),(3,6),(4,7),(5,8),(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40)], [(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)], [(1,23),(2,22),(3,21),(4,25),(5,24),(6,26),(7,30),(8,29),(9,28),(10,27),(11,31),(12,35),(13,34),(14,33),(15,32),(16,36),(17,40),(18,39),(19,38),(20,37)]])
C23×D5 is a maximal subgroup of
C22⋊D20 C23⋊D10
C23×D5 is a maximal quotient of D4⋊6D10 Q8.10D10 D4⋊8D10 D4.10D10
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 5A | 5B | 10A | ··· | 10N |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 5 | 5 | 10 | ··· | 10 |
| size | 1 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | 2 | 2 | ··· | 2 |
32 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | C2 | D5 | D10 |
| kernel | C23×D5 | C22×D5 | C22×C10 | C23 | C22 |
| # reps | 1 | 14 | 1 | 2 | 14 |
Matrix representation of C23×D5 ►in GL4(𝔽11) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 10 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 1 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 10 | 3 |
| 10 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 10 |
| 0 | 0 | 10 | 0 |
G:=sub<GL(4,GF(11))| [1,0,0,0,0,1,0,0,0,0,10,0,0,0,0,10],[10,0,0,0,0,1,0,0,0,0,10,0,0,0,0,10],[1,0,0,0,0,10,0,0,0,0,10,0,0,0,0,10],[1,0,0,0,0,1,0,0,0,0,0,10,0,0,1,3],[10,0,0,0,0,10,0,0,0,0,0,10,0,0,10,0] >;
C23×D5 in GAP, Magma, Sage, TeX
C_2^3\times D_5
% in TeX
G:=Group("C2^3xD5"); // GroupNames label
G:=SmallGroup(80,51);
// by ID
G=gap.SmallGroup(80,51);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-5,1604]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^5=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations