direct product, metabelian, soluble, monomial, A-group
Aliases: C2×C52⋊C9, C52⋊2C18, (C5×C10)⋊C9, (C5×C30).C3, C6.(C52⋊C3), (C5×C15).2C6, C3.(C2×C52⋊C3), SmallGroup(450,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C2×C52⋊C9 |
Generators and relations for C2×C52⋊C9
G = < a,b,c,d | a2=b5=c5=d9=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b3c2 >
Smallest permutation representation of C2×C52⋊C9
►On 90 points
Generators in S90
(1 32)(2 33)(3 34)(4 35)(5 36)(6 28)(7 29)(8 30)(9 31)(10 53)(11 54)(12 46)(13 47)(14 48)(15 49)(16 50)(17 51)(18 52)(19 62)(20 63)(21 55)(22 56)(23 57)(24 58)(25 59)(26 60)(27 61)(37 80)(38 81)(39 73)(40 74)(41 75)(42 76)(43 77)(44 78)(45 79)(64 89)(65 90)(66 82)(67 83)(68 84)(69 85)(70 86)(71 87)(72 88)
(2 75 69 52 59)(3 60 53 70 76)(5 78 72 46 62)(6 63 47 64 79)(8 81 66 49 56)(9 57 50 67 73)(10 86 42 34 26)(12 19 36 44 88)(13 89 45 28 20)(15 22 30 38 82)(16 83 39 31 23)(18 25 33 41 85)
(1 51 74 58 68)(2 52 75 59 69)(3 60 53 70 76)(4 54 77 61 71)(5 46 78 62 72)(6 63 47 64 79)(7 48 80 55 65)(8 49 81 56 66)(9 57 50 67 73)(10 86 42 34 26)(11 43 27 87 35)(12 44 19 88 36)(13 89 45 28 20)(14 37 21 90 29)(15 38 22 82 30)(16 83 39 31 23)(17 40 24 84 32)(18 41 25 85 33)
(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)
G:=sub<Sym(90)| (1,32)(2,33)(3,34)(4,35)(5,36)(6,28)(7,29)(8,30)(9,31)(10,53)(11,54)(12,46)(13,47)(14,48)(15,49)(16,50)(17,51)(18,52)(19,62)(20,63)(21,55)(22,56)(23,57)(24,58)(25,59)(26,60)(27,61)(37,80)(38,81)(39,73)(40,74)(41,75)(42,76)(43,77)(44,78)(45,79)(64,89)(65,90)(66,82)(67,83)(68,84)(69,85)(70,86)(71,87)(72,88), (2,75,69,52,59)(3,60,53,70,76)(5,78,72,46,62)(6,63,47,64,79)(8,81,66,49,56)(9,57,50,67,73)(10,86,42,34,26)(12,19,36,44,88)(13,89,45,28,20)(15,22,30,38,82)(16,83,39,31,23)(18,25,33,41,85), (1,51,74,58,68)(2,52,75,59,69)(3,60,53,70,76)(4,54,77,61,71)(5,46,78,62,72)(6,63,47,64,79)(7,48,80,55,65)(8,49,81,56,66)(9,57,50,67,73)(10,86,42,34,26)(11,43,27,87,35)(12,44,19,88,36)(13,89,45,28,20)(14,37,21,90,29)(15,38,22,82,30)(16,83,39,31,23)(17,40,24,84,32)(18,41,25,85,33), (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)>;
G:=Group( (1,32)(2,33)(3,34)(4,35)(5,36)(6,28)(7,29)(8,30)(9,31)(10,53)(11,54)(12,46)(13,47)(14,48)(15,49)(16,50)(17,51)(18,52)(19,62)(20,63)(21,55)(22,56)(23,57)(24,58)(25,59)(26,60)(27,61)(37,80)(38,81)(39,73)(40,74)(41,75)(42,76)(43,77)(44,78)(45,79)(64,89)(65,90)(66,82)(67,83)(68,84)(69,85)(70,86)(71,87)(72,88), (2,75,69,52,59)(3,60,53,70,76)(5,78,72,46,62)(6,63,47,64,79)(8,81,66,49,56)(9,57,50,67,73)(10,86,42,34,26)(12,19,36,44,88)(13,89,45,28,20)(15,22,30,38,82)(16,83,39,31,23)(18,25,33,41,85), (1,51,74,58,68)(2,52,75,59,69)(3,60,53,70,76)(4,54,77,61,71)(5,46,78,62,72)(6,63,47,64,79)(7,48,80,55,65)(8,49,81,56,66)(9,57,50,67,73)(10,86,42,34,26)(11,43,27,87,35)(12,44,19,88,36)(13,89,45,28,20)(14,37,21,90,29)(15,38,22,82,30)(16,83,39,31,23)(17,40,24,84,32)(18,41,25,85,33), (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) );
G=PermutationGroup([[(1,32),(2,33),(3,34),(4,35),(5,36),(6,28),(7,29),(8,30),(9,31),(10,53),(11,54),(12,46),(13,47),(14,48),(15,49),(16,50),(17,51),(18,52),(19,62),(20,63),(21,55),(22,56),(23,57),(24,58),(25,59),(26,60),(27,61),(37,80),(38,81),(39,73),(40,74),(41,75),(42,76),(43,77),(44,78),(45,79),(64,89),(65,90),(66,82),(67,83),(68,84),(69,85),(70,86),(71,87),(72,88)], [(2,75,69,52,59),(3,60,53,70,76),(5,78,72,46,62),(6,63,47,64,79),(8,81,66,49,56),(9,57,50,67,73),(10,86,42,34,26),(12,19,36,44,88),(13,89,45,28,20),(15,22,30,38,82),(16,83,39,31,23),(18,25,33,41,85)], [(1,51,74,58,68),(2,52,75,59,69),(3,60,53,70,76),(4,54,77,61,71),(5,46,78,62,72),(6,63,47,64,79),(7,48,80,55,65),(8,49,81,56,66),(9,57,50,67,73),(10,86,42,34,26),(11,43,27,87,35),(12,44,19,88,36),(13,89,45,28,20),(14,37,21,90,29),(15,38,22,82,30),(16,83,39,31,23),(17,40,24,84,32),(18,41,25,85,33)], [(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)]])
66 conjugacy classes
| class | 1 | 2 | 3A | 3B | 5A | ··· | 5H | 6A | 6B | 9A | ··· | 9F | 10A | ··· | 10H | 15A | ··· | 15P | 18A | ··· | 18F | 30A | ··· | 30P |
| order | 1 | 2 | 3 | 3 | 5 | ··· | 5 | 6 | 6 | 9 | ··· | 9 | 10 | ··· | 10 | 15 | ··· | 15 | 18 | ··· | 18 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 1 | 1 | 25 | ··· | 25 | 3 | ··· | 3 | 3 | ··· | 3 | 25 | ··· | 25 | 3 | ··· | 3 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | C52⋊C3 | C2×C52⋊C3 | C52⋊C9 | C2×C52⋊C9 |
| kernel | C2×C52⋊C9 | C52⋊C9 | C5×C30 | C5×C15 | C5×C10 | C52 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 8 | 8 | 16 | 16 |
Matrix representation of C2×C52⋊C9 ►in GL4(𝔽181) generated by
| 180 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 59 | 0 |
| 0 | 0 | 0 | 135 |
| 1 | 0 | 0 | 0 |
| 0 | 42 | 0 | 0 |
| 0 | 0 | 59 | 0 |
| 0 | 0 | 0 | 42 |
| 132 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 132 | 0 | 0 |
G:=sub<GL(4,GF(181))| [180,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,59,0,0,0,0,135],[1,0,0,0,0,42,0,0,0,0,59,0,0,0,0,42],[132,0,0,0,0,0,0,132,0,1,0,0,0,0,1,0] >;
C2×C52⋊C9 in GAP, Magma, Sage, TeX
C_2\times C_5^2\rtimes C_9
% in TeX
G:=Group("C2xC5^2:C9"); // GroupNames label
G:=SmallGroup(450,13);
// by ID
G=gap.SmallGroup(450,13);
# by ID
G:=PCGroup([5,-2,-3,-3,-5,5,36,2888,4284]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^5=c^5=d^9=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,d*b*d^-1=b^3*c^2>;
// generators/relations
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