direct product, metacyclic, supersoluble, monomial, Z-group, 5-hyperelementary
Aliases: C9×C11⋊C5, C99⋊C5, C11⋊C45, C33.C15, C3.(C3×C11⋊C5), (C3×C11⋊C5).C3, SmallGroup(495,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C9×C11⋊C5 |
Generators and relations for C9×C11⋊C5
G = < a,b,c | a9=b11=c5=1, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of C9×C11⋊C5
►On 99 points
Generators in S99
(1 89 56 23 78 45 12 67 34)(2 90 57 24 79 46 13 68 35)(3 91 58 25 80 47 14 69 36)(4 92 59 26 81 48 15 70 37)(5 93 60 27 82 49 16 71 38)(6 94 61 28 83 50 17 72 39)(7 95 62 29 84 51 18 73 40)(8 96 63 30 85 52 19 74 41)(9 97 64 31 86 53 20 75 42)(10 98 65 32 87 54 21 76 43)(11 99 66 33 88 55 22 77 44)
(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 81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96 97 98 99)
(2 5 6 10 4)(3 9 11 8 7)(13 16 17 21 15)(14 20 22 19 18)(24 27 28 32 26)(25 31 33 30 29)(35 38 39 43 37)(36 42 44 41 40)(46 49 50 54 48)(47 53 55 52 51)(57 60 61 65 59)(58 64 66 63 62)(68 71 72 76 70)(69 75 77 74 73)(79 82 83 87 81)(80 86 88 85 84)(90 93 94 98 92)(91 97 99 96 95)
G:=sub<Sym(99)| (1,89,56,23,78,45,12,67,34)(2,90,57,24,79,46,13,68,35)(3,91,58,25,80,47,14,69,36)(4,92,59,26,81,48,15,70,37)(5,93,60,27,82,49,16,71,38)(6,94,61,28,83,50,17,72,39)(7,95,62,29,84,51,18,73,40)(8,96,63,30,85,52,19,74,41)(9,97,64,31,86,53,20,75,42)(10,98,65,32,87,54,21,76,43)(11,99,66,33,88,55,22,77,44), (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)(78,79,80,81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96,97,98,99), (2,5,6,10,4)(3,9,11,8,7)(13,16,17,21,15)(14,20,22,19,18)(24,27,28,32,26)(25,31,33,30,29)(35,38,39,43,37)(36,42,44,41,40)(46,49,50,54,48)(47,53,55,52,51)(57,60,61,65,59)(58,64,66,63,62)(68,71,72,76,70)(69,75,77,74,73)(79,82,83,87,81)(80,86,88,85,84)(90,93,94,98,92)(91,97,99,96,95)>;
G:=Group( (1,89,56,23,78,45,12,67,34)(2,90,57,24,79,46,13,68,35)(3,91,58,25,80,47,14,69,36)(4,92,59,26,81,48,15,70,37)(5,93,60,27,82,49,16,71,38)(6,94,61,28,83,50,17,72,39)(7,95,62,29,84,51,18,73,40)(8,96,63,30,85,52,19,74,41)(9,97,64,31,86,53,20,75,42)(10,98,65,32,87,54,21,76,43)(11,99,66,33,88,55,22,77,44), (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)(78,79,80,81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96,97,98,99), (2,5,6,10,4)(3,9,11,8,7)(13,16,17,21,15)(14,20,22,19,18)(24,27,28,32,26)(25,31,33,30,29)(35,38,39,43,37)(36,42,44,41,40)(46,49,50,54,48)(47,53,55,52,51)(57,60,61,65,59)(58,64,66,63,62)(68,71,72,76,70)(69,75,77,74,73)(79,82,83,87,81)(80,86,88,85,84)(90,93,94,98,92)(91,97,99,96,95) );
G=PermutationGroup([[(1,89,56,23,78,45,12,67,34),(2,90,57,24,79,46,13,68,35),(3,91,58,25,80,47,14,69,36),(4,92,59,26,81,48,15,70,37),(5,93,60,27,82,49,16,71,38),(6,94,61,28,83,50,17,72,39),(7,95,62,29,84,51,18,73,40),(8,96,63,30,85,52,19,74,41),(9,97,64,31,86,53,20,75,42),(10,98,65,32,87,54,21,76,43),(11,99,66,33,88,55,22,77,44)], [(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),(78,79,80,81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96,97,98,99)], [(2,5,6,10,4),(3,9,11,8,7),(13,16,17,21,15),(14,20,22,19,18),(24,27,28,32,26),(25,31,33,30,29),(35,38,39,43,37),(36,42,44,41,40),(46,49,50,54,48),(47,53,55,52,51),(57,60,61,65,59),(58,64,66,63,62),(68,71,72,76,70),(69,75,77,74,73),(79,82,83,87,81),(80,86,88,85,84),(90,93,94,98,92),(91,97,99,96,95)]])
63 conjugacy classes
| class | 1 | 3A | 3B | 5A | 5B | 5C | 5D | 9A | ··· | 9F | 11A | 11B | 15A | ··· | 15H | 33A | 33B | 33C | 33D | 45A | ··· | 45X | 99A | ··· | 99L |
| order | 1 | 3 | 3 | 5 | 5 | 5 | 5 | 9 | ··· | 9 | 11 | 11 | 15 | ··· | 15 | 33 | 33 | 33 | 33 | 45 | ··· | 45 | 99 | ··· | 99 |
| size | 1 | 1 | 1 | 11 | 11 | 11 | 11 | 1 | ··· | 1 | 5 | 5 | 11 | ··· | 11 | 5 | 5 | 5 | 5 | 11 | ··· | 11 | 5 | ··· | 5 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 5 | 5 | 5 |
| type | + | ||||||||
| image | C1 | C3 | C5 | C9 | C15 | C45 | C11⋊C5 | C3×C11⋊C5 | C9×C11⋊C5 |
| kernel | C9×C11⋊C5 | C3×C11⋊C5 | C99 | C11⋊C5 | C33 | C11 | C9 | C3 | C1 |
| # reps | 1 | 2 | 4 | 6 | 8 | 24 | 2 | 4 | 12 |
Matrix representation of C9×C11⋊C5 ►in GL5(𝔽991)
| 18 | 0 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 0 | 18 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 937 |
| 0 | 1 | 0 | 0 | 990 |
| 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 0 | 1 | 936 |
| 53 | 1 | 0 | 2 | 0 |
| 56 | 0 | 0 | 936 | 1 |
| 936 | 0 | 0 | 54 | 0 |
| 54 | 0 | 0 | 938 | 0 |
| 2 | 0 | 1 | 935 | 0 |
G:=sub<GL(5,GF(991))| [18,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18],[0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,1,937,990,1,936],[53,56,936,54,2,1,0,0,0,0,0,0,0,0,1,2,936,54,938,935,0,1,0,0,0] >;
C9×C11⋊C5 in GAP, Magma, Sage, TeX
C_9\times C_{11}\rtimes C_5 % in TeX
G:=Group("C9xC11:C5"); // GroupNames label
G:=SmallGroup(495,1);
// by ID
G=gap.SmallGroup(495,1);
# by ID
G:=PCGroup([4,-3,-5,-3,-11,60,967]);
// Polycyclic
G:=Group<a,b,c|a^9=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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