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G = C9×C11⋊C5order 495 = 32·5·11

Direct product of C9 and C11⋊C5

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

C1C11 — C9×C11⋊C5
C1C11C33C3×C11⋊C5 — C9×C11⋊C5
C11 — C9×C11⋊C5
C1C9

Generators and relations for C9×C11⋊C5
 G = < a,b,c | a9=b11=c5=1, ab=ba, ac=ca, cbc-1=b3 >

11C5
11C15
11C45

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 3A3B5A5B5C5D9A···9F11A11B15A···15H33A33B33C33D45A···45X99A···99L
order13355559···9111115···153333333345···4599···99
size111111111111···15511···11555511···115···5

63 irreducible representations

dim111111555
type+
imageC1C3C5C9C15C45C11⋊C5C3×C11⋊C5C9×C11⋊C5
kernelC9×C11⋊C5C3×C11⋊C5C99C11⋊C5C33C11C9C3C1
# reps12468242412

Matrix representation of C9×C11⋊C5 in GL5(𝔽991)

180000
018000
001800
000180
000018
,
00001
1000937
0100990
00101
0001936
,
531020
56009361
93600540
54009380
2019350

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

Export

Subgroup lattice of C9×C11⋊C5 in TeX

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