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## G = C32×C11⋊C5order 495 = 32·5·11

### Direct product of C32 and C11⋊C5

Aliases: C32×C11⋊C5, C33⋊C15, (C3×C33)⋊C5, C11⋊(C3×C15), SmallGroup(495,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C11 — C32×C11⋊C5
 Chief series C1 — C11 — C11⋊C5 — C3×C11⋊C5 — C32×C11⋊C5
 Lower central C11 — C32×C11⋊C5
 Upper central C1 — C32

Generators and relations for C32×C11⋊C5
G = < a,b,c,d | a3=b3=c11=d5=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Smallest permutation representation of C32×C11⋊C5
On 99 points
Generators in S99
(1 89 45)(2 90 46)(3 91 47)(4 92 48)(5 93 49)(6 94 50)(7 95 51)(8 96 52)(9 97 53)(10 98 54)(11 99 55)(12 67 56)(13 68 57)(14 69 58)(15 70 59)(16 71 60)(17 72 61)(18 73 62)(19 74 63)(20 75 64)(21 76 65)(22 77 66)(23 78 34)(24 79 35)(25 80 36)(26 81 37)(27 82 38)(28 83 39)(29 84 40)(30 85 41)(31 86 42)(32 87 43)(33 88 44)
(1 23 12)(2 24 13)(3 25 14)(4 26 15)(5 27 16)(6 28 17)(7 29 18)(8 30 19)(9 31 20)(10 32 21)(11 33 22)(34 56 45)(35 57 46)(36 58 47)(37 59 48)(38 60 49)(39 61 50)(40 62 51)(41 63 52)(42 64 53)(43 65 54)(44 66 55)(67 89 78)(68 90 79)(69 91 80)(70 92 81)(71 93 82)(72 94 83)(73 95 84)(74 96 85)(75 97 86)(76 98 87)(77 99 88)
(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)| (1,89,45)(2,90,46)(3,91,47)(4,92,48)(5,93,49)(6,94,50)(7,95,51)(8,96,52)(9,97,53)(10,98,54)(11,99,55)(12,67,56)(13,68,57)(14,69,58)(15,70,59)(16,71,60)(17,72,61)(18,73,62)(19,74,63)(20,75,64)(21,76,65)(22,77,66)(23,78,34)(24,79,35)(25,80,36)(26,81,37)(27,82,38)(28,83,39)(29,84,40)(30,85,41)(31,86,42)(32,87,43)(33,88,44), (1,23,12)(2,24,13)(3,25,14)(4,26,15)(5,27,16)(6,28,17)(7,29,18)(8,30,19)(9,31,20)(10,32,21)(11,33,22)(34,56,45)(35,57,46)(36,58,47)(37,59,48)(38,60,49)(39,61,50)(40,62,51)(41,63,52)(42,64,53)(43,65,54)(44,66,55)(67,89,78)(68,90,79)(69,91,80)(70,92,81)(71,93,82)(72,94,83)(73,95,84)(74,96,85)(75,97,86)(76,98,87)(77,99,88), (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( (1,89,45)(2,90,46)(3,91,47)(4,92,48)(5,93,49)(6,94,50)(7,95,51)(8,96,52)(9,97,53)(10,98,54)(11,99,55)(12,67,56)(13,68,57)(14,69,58)(15,70,59)(16,71,60)(17,72,61)(18,73,62)(19,74,63)(20,75,64)(21,76,65)(22,77,66)(23,78,34)(24,79,35)(25,80,36)(26,81,37)(27,82,38)(28,83,39)(29,84,40)(30,85,41)(31,86,42)(32,87,43)(33,88,44), (1,23,12)(2,24,13)(3,25,14)(4,26,15)(5,27,16)(6,28,17)(7,29,18)(8,30,19)(9,31,20)(10,32,21)(11,33,22)(34,56,45)(35,57,46)(36,58,47)(37,59,48)(38,60,49)(39,61,50)(40,62,51)(41,63,52)(42,64,53)(43,65,54)(44,66,55)(67,89,78)(68,90,79)(69,91,80)(70,92,81)(71,93,82)(72,94,83)(73,95,84)(74,96,85)(75,97,86)(76,98,87)(77,99,88), (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([(1,89,45),(2,90,46),(3,91,47),(4,92,48),(5,93,49),(6,94,50),(7,95,51),(8,96,52),(9,97,53),(10,98,54),(11,99,55),(12,67,56),(13,68,57),(14,69,58),(15,70,59),(16,71,60),(17,72,61),(18,73,62),(19,74,63),(20,75,64),(21,76,65),(22,77,66),(23,78,34),(24,79,35),(25,80,36),(26,81,37),(27,82,38),(28,83,39),(29,84,40),(30,85,41),(31,86,42),(32,87,43),(33,88,44)], [(1,23,12),(2,24,13),(3,25,14),(4,26,15),(5,27,16),(6,28,17),(7,29,18),(8,30,19),(9,31,20),(10,32,21),(11,33,22),(34,56,45),(35,57,46),(36,58,47),(37,59,48),(38,60,49),(39,61,50),(40,62,51),(41,63,52),(42,64,53),(43,65,54),(44,66,55),(67,89,78),(68,90,79),(69,91,80),(70,92,81),(71,93,82),(72,94,83),(73,95,84),(74,96,85),(75,97,86),(76,98,87),(77,99,88)], [(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 ··· 3H 5A 5B 5C 5D 11A 11B 15A ··· 15AF 33A ··· 33P order 1 3 ··· 3 5 5 5 5 11 11 15 ··· 15 33 ··· 33 size 1 1 ··· 1 11 11 11 11 5 5 11 ··· 11 5 ··· 5

63 irreducible representations

 dim 1 1 1 1 5 5 type + image C1 C3 C5 C15 C11⋊C5 C3×C11⋊C5 kernel C32×C11⋊C5 C3×C11⋊C5 C3×C33 C33 C32 C3 # reps 1 8 4 32 2 16

Matrix representation of C32×C11⋊C5 in GL6(𝔽331)

 31 0 0 0 0 0 0 31 0 0 0 0 0 0 31 0 0 0 0 0 0 31 0 0 0 0 0 0 31 0 0 0 0 0 0 31
,
 1 0 0 0 0 0 0 299 0 0 0 0 0 0 299 0 0 0 0 0 0 299 0 0 0 0 0 0 299 0 0 0 0 0 0 299
,
 1 0 0 0 0 0 0 104 1 330 105 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 150 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 226 329 1 225 103 0 225 103 1 329 104 0 0 1 0 0 0

G:=sub<GL(6,GF(331))| [31,0,0,0,0,0,0,31,0,0,0,0,0,0,31,0,0,0,0,0,0,31,0,0,0,0,0,0,31,0,0,0,0,0,0,31],[1,0,0,0,0,0,0,299,0,0,0,0,0,0,299,0,0,0,0,0,0,299,0,0,0,0,0,0,299,0,0,0,0,0,0,299],[1,0,0,0,0,0,0,104,1,0,0,0,0,1,0,1,0,0,0,330,0,0,1,0,0,105,0,0,0,1,0,1,0,0,0,0],[150,0,0,0,0,0,0,1,0,226,225,0,0,0,0,329,103,1,0,0,0,1,1,0,0,0,1,225,329,0,0,0,0,103,104,0] >;

C32×C11⋊C5 in GAP, Magma, Sage, TeX

C_3^2\times C_{11}\rtimes C_5
% in TeX

G:=Group("C3^2xC11:C5");
// GroupNames label

G:=SmallGroup(495,3);
// by ID

G=gap.SmallGroup(495,3);
# by ID

G:=PCGroup([4,-3,-3,-5,-11,331]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^11=d^5=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

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