metabelian, soluble, monomial, A-group
Aliases: C24⋊C25, C5.(C24⋊C5), (C23×C10).C5, SmallGroup(400,52)
Series: Derived ►Chief ►Lower central ►Upper central
| C24 — C24⋊C25 |
Generators and relations for C24⋊C25
G = < a,b,c,d,e | a2=b2=c2=d2=e25=1, ab=ba, ac=ca, ad=da, eae-1=abc, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, ede-1=a >
Smallest permutation representation of C24⋊C25
►On 50 points
Generators in S50
(2 49)(3 50)(4 26)(5 27)(7 29)(8 30)(9 31)(10 32)(12 34)(13 35)(14 36)(15 37)(17 39)(18 40)(19 41)(20 42)(22 44)(23 45)(24 46)(25 47)
(4 26)(5 27)(9 31)(10 32)(14 36)(15 37)(19 41)(20 42)(24 46)(25 47)
(1 48)(4 26)(6 28)(9 31)(11 33)(14 36)(16 38)(19 41)(21 43)(24 46)
(1 48)(3 50)(4 26)(5 27)(6 28)(8 30)(9 31)(10 32)(11 33)(13 35)(14 36)(15 37)(16 38)(18 40)(19 41)(20 42)(21 43)(23 45)(24 46)(25 47)
(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)
G:=sub<Sym(50)| (2,49)(3,50)(4,26)(5,27)(7,29)(8,30)(9,31)(10,32)(12,34)(13,35)(14,36)(15,37)(17,39)(18,40)(19,41)(20,42)(22,44)(23,45)(24,46)(25,47), (4,26)(5,27)(9,31)(10,32)(14,36)(15,37)(19,41)(20,42)(24,46)(25,47), (1,48)(4,26)(6,28)(9,31)(11,33)(14,36)(16,38)(19,41)(21,43)(24,46), (1,48)(3,50)(4,26)(5,27)(6,28)(8,30)(9,31)(10,32)(11,33)(13,35)(14,36)(15,37)(16,38)(18,40)(19,41)(20,42)(21,43)(23,45)(24,46)(25,47), (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)>;
G:=Group( (2,49)(3,50)(4,26)(5,27)(7,29)(8,30)(9,31)(10,32)(12,34)(13,35)(14,36)(15,37)(17,39)(18,40)(19,41)(20,42)(22,44)(23,45)(24,46)(25,47), (4,26)(5,27)(9,31)(10,32)(14,36)(15,37)(19,41)(20,42)(24,46)(25,47), (1,48)(4,26)(6,28)(9,31)(11,33)(14,36)(16,38)(19,41)(21,43)(24,46), (1,48)(3,50)(4,26)(5,27)(6,28)(8,30)(9,31)(10,32)(11,33)(13,35)(14,36)(15,37)(16,38)(18,40)(19,41)(20,42)(21,43)(23,45)(24,46)(25,47), (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) );
G=PermutationGroup([[(2,49),(3,50),(4,26),(5,27),(7,29),(8,30),(9,31),(10,32),(12,34),(13,35),(14,36),(15,37),(17,39),(18,40),(19,41),(20,42),(22,44),(23,45),(24,46),(25,47)], [(4,26),(5,27),(9,31),(10,32),(14,36),(15,37),(19,41),(20,42),(24,46),(25,47)], [(1,48),(4,26),(6,28),(9,31),(11,33),(14,36),(16,38),(19,41),(21,43),(24,46)], [(1,48),(3,50),(4,26),(5,27),(6,28),(8,30),(9,31),(10,32),(11,33),(13,35),(14,36),(15,37),(16,38),(18,40),(19,41),(20,42),(21,43),(23,45),(24,46),(25,47)], [(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)]])
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 25A | ··· | 25T |
| order | 1 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 25 | ··· | 25 |
| size | 1 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | 5 | ··· | 5 | 16 | ··· | 16 |
40 irreducible representations
| dim | 1 | 1 | 1 | 5 | 5 |
| type | + | + | |||
| image | C1 | C5 | C25 | C24⋊C5 | C24⋊C25 |
| kernel | C24⋊C25 | C23×C10 | C24 | C5 | C1 |
| # reps | 1 | 4 | 20 | 3 | 12 |
Matrix representation of C24⋊C25 ►in GL5(𝔽101)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 100 | 0 | 0 | 0 |
| 33 | 0 | 100 | 0 | 0 |
| 70 | 0 | 0 | 100 | 0 |
| 1 | 0 | 0 | 0 | 100 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 100 | 0 | 0 | 0 |
| 33 | 0 | 100 | 0 | 0 |
| 0 | 52 | 0 | 1 | 0 |
| 0 | 25 | 0 | 0 | 1 |
| 100 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 97 | 100 | 0 | 0 |
| 31 | 0 | 0 | 1 | 0 |
| 100 | 0 | 0 | 0 | 1 |
| 100 | 0 | 0 | 0 | 0 |
| 0 | 100 | 0 | 0 | 0 |
| 0 | 0 | 100 | 0 | 0 |
| 0 | 0 | 0 | 100 | 0 |
| 100 | 25 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 |
| 33 | 97 | 99 | 0 | 0 |
| 0 | 0 | 4 | 1 | 0 |
| 0 | 0 | 52 | 0 | 1 |
| 0 | 0 | 25 | 0 | 0 |
G:=sub<GL(5,GF(101))| [1,0,33,70,1,0,100,0,0,0,0,0,100,0,0,0,0,0,100,0,0,0,0,0,100],[1,0,33,0,0,0,100,0,52,25,0,0,100,0,0,0,0,0,1,0,0,0,0,0,1],[100,0,0,31,100,0,1,97,0,0,0,0,100,0,0,0,0,0,1,0,0,0,0,0,1],[100,0,0,0,100,0,100,0,0,25,0,0,100,0,0,0,0,0,100,0,0,0,0,0,1],[0,33,0,0,0,1,97,0,0,0,0,99,4,52,25,0,0,1,0,0,0,0,0,1,0] >;
C24⋊C25 in GAP, Magma, Sage, TeX
C_2^4\rtimes C_{25} % in TeX
G:=Group("C2^4:C25"); // GroupNames label
G:=SmallGroup(400,52);
// by ID
G=gap.SmallGroup(400,52);
# by ID
G:=PCGroup([6,-5,-5,-2,2,2,2,30,3602,5403,8254,13505]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=e^25=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b*c,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,e*c*e^-1=c*d=d*c,e*d*e^-1=a>;
// generators/relations
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