direct product, metacyclic, supersoluble, monomial, A-group, 5-hyperelementary
Aliases: C5×C11⋊C5, C55⋊C5, C11⋊C52, SmallGroup(275,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C5×C11⋊C5 |
Generators and relations for C5×C11⋊C5
G = < a,b,c | a5=b11=c5=1, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of C5×C11⋊C5
►On 55 points
Generators in S55
(1 45 34 23 12)(2 46 35 24 13)(3 47 36 25 14)(4 48 37 26 15)(5 49 38 27 16)(6 50 39 28 17)(7 51 40 29 18)(8 52 41 30 19)(9 53 42 31 20)(10 54 43 32 21)(11 55 44 33 22)
(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)
(1 23 45 12 34)(2 27 50 21 37)(3 31 55 19 40)(4 24 49 17 43)(5 28 54 15 35)(6 32 48 13 38)(7 25 53 22 41)(8 29 47 20 44)(9 33 52 18 36)(10 26 46 16 39)(11 30 51 14 42)
G:=sub<Sym(55)| (1,45,34,23,12)(2,46,35,24,13)(3,47,36,25,14)(4,48,37,26,15)(5,49,38,27,16)(6,50,39,28,17)(7,51,40,29,18)(8,52,41,30,19)(9,53,42,31,20)(10,54,43,32,21)(11,55,44,33,22), (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), (1,23,45,12,34)(2,27,50,21,37)(3,31,55,19,40)(4,24,49,17,43)(5,28,54,15,35)(6,32,48,13,38)(7,25,53,22,41)(8,29,47,20,44)(9,33,52,18,36)(10,26,46,16,39)(11,30,51,14,42)>;
G:=Group( (1,45,34,23,12)(2,46,35,24,13)(3,47,36,25,14)(4,48,37,26,15)(5,49,38,27,16)(6,50,39,28,17)(7,51,40,29,18)(8,52,41,30,19)(9,53,42,31,20)(10,54,43,32,21)(11,55,44,33,22), (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), (1,23,45,12,34)(2,27,50,21,37)(3,31,55,19,40)(4,24,49,17,43)(5,28,54,15,35)(6,32,48,13,38)(7,25,53,22,41)(8,29,47,20,44)(9,33,52,18,36)(10,26,46,16,39)(11,30,51,14,42) );
G=PermutationGroup([[(1,45,34,23,12),(2,46,35,24,13),(3,47,36,25,14),(4,48,37,26,15),(5,49,38,27,16),(6,50,39,28,17),(7,51,40,29,18),(8,52,41,30,19),(9,53,42,31,20),(10,54,43,32,21),(11,55,44,33,22)], [(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)], [(1,23,45,12,34),(2,27,50,21,37),(3,31,55,19,40),(4,24,49,17,43),(5,28,54,15,35),(6,32,48,13,38),(7,25,53,22,41),(8,29,47,20,44),(9,33,52,18,36),(10,26,46,16,39),(11,30,51,14,42)]])
35 conjugacy classes
| class | 1 | 5A | 5B | 5C | 5D | 5E | ··· | 5X | 11A | 11B | 55A | ··· | 55H |
| order | 1 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 11 | 11 | 55 | ··· | 55 |
| size | 1 | 1 | 1 | 1 | 1 | 11 | ··· | 11 | 5 | 5 | 5 | ··· | 5 |
35 irreducible representations
| dim | 1 | 1 | 1 | 5 | 5 |
| type | + | ||||
| image | C1 | C5 | C5 | C11⋊C5 | C5×C11⋊C5 |
| kernel | C5×C11⋊C5 | C11⋊C5 | C55 | C5 | C1 |
| # reps | 1 | 20 | 4 | 2 | 8 |
Matrix representation of C5×C11⋊C5 ►in GL6(𝔽331)
| 124 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 225 | 2 | 103 | 226 | 1 |
| 0 | 226 | 2 | 103 | 226 | 1 |
| 0 | 225 | 3 | 103 | 226 | 1 |
| 0 | 225 | 2 | 104 | 226 | 1 |
| 0 | 225 | 2 | 103 | 227 | 1 |
| 323 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 227 | 106 | 329 | 228 | 105 |
| 0 | 228 | 105 | 225 | 2 | 104 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(331))| [124,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,225,226,225,225,225,0,2,2,3,2,2,0,103,103,103,104,103,0,226,226,226,226,227,0,1,1,1,1,1],[323,0,0,0,0,0,0,0,227,228,1,0,0,0,106,105,0,0,0,1,329,225,0,0,0,0,228,2,0,1,0,0,105,104,0,0] >;
C5×C11⋊C5 in GAP, Magma, Sage, TeX
C_5\times C_{11}\rtimes C_5 % in TeX
G:=Group("C5xC11:C5"); // GroupNames label
G:=SmallGroup(275,3);
// by ID
G=gap.SmallGroup(275,3);
# by ID
G:=PCGroup([3,-5,-5,-11,902]);
// Polycyclic
G:=Group<a,b,c|a^5=b^11=c^5=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
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