metabelian, supersoluble, monomial, A-group
Aliases: C3⋊D33, C33⋊1S3, C32⋊2D11, C11⋊(C3⋊S3), (C3×C33)⋊1C2, SmallGroup(198,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C33 — C3⋊D33 |
Generators and relations for C3⋊D33
G = < a,b,c | a3=b33=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Smallest permutation representation of C3⋊D33
►On 99 points
Generators in S99
(1 92 65)(2 93 66)(3 94 34)(4 95 35)(5 96 36)(6 97 37)(7 98 38)(8 99 39)(9 67 40)(10 68 41)(11 69 42)(12 70 43)(13 71 44)(14 72 45)(15 73 46)(16 74 47)(17 75 48)(18 76 49)(19 77 50)(20 78 51)(21 79 52)(22 80 53)(23 81 54)(24 82 55)(25 83 56)(26 84 57)(27 85 58)(28 86 59)(29 87 60)(30 88 61)(31 89 62)(32 90 63)(33 91 64)
(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)
(1 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(34 89)(35 88)(36 87)(37 86)(38 85)(39 84)(40 83)(41 82)(42 81)(43 80)(44 79)(45 78)(46 77)(47 76)(48 75)(49 74)(50 73)(51 72)(52 71)(53 70)(54 69)(55 68)(56 67)(57 99)(58 98)(59 97)(60 96)(61 95)(62 94)(63 93)(64 92)(65 91)(66 90)
G:=sub<Sym(99)| (1,92,65)(2,93,66)(3,94,34)(4,95,35)(5,96,36)(6,97,37)(7,98,38)(8,99,39)(9,67,40)(10,68,41)(11,69,42)(12,70,43)(13,71,44)(14,72,45)(15,73,46)(16,74,47)(17,75,48)(18,76,49)(19,77,50)(20,78,51)(21,79,52)(22,80,53)(23,81,54)(24,82,55)(25,83,56)(26,84,57)(27,85,58)(28,86,59)(29,87,60)(30,88,61)(31,89,62)(32,90,63)(33,91,64), (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,89)(35,88)(36,87)(37,86)(38,85)(39,84)(40,83)(41,82)(42,81)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,99)(58,98)(59,97)(60,96)(61,95)(62,94)(63,93)(64,92)(65,91)(66,90)>;
G:=Group( (1,92,65)(2,93,66)(3,94,34)(4,95,35)(5,96,36)(6,97,37)(7,98,38)(8,99,39)(9,67,40)(10,68,41)(11,69,42)(12,70,43)(13,71,44)(14,72,45)(15,73,46)(16,74,47)(17,75,48)(18,76,49)(19,77,50)(20,78,51)(21,79,52)(22,80,53)(23,81,54)(24,82,55)(25,83,56)(26,84,57)(27,85,58)(28,86,59)(29,87,60)(30,88,61)(31,89,62)(32,90,63)(33,91,64), (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,89)(35,88)(36,87)(37,86)(38,85)(39,84)(40,83)(41,82)(42,81)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,99)(58,98)(59,97)(60,96)(61,95)(62,94)(63,93)(64,92)(65,91)(66,90) );
G=PermutationGroup([[(1,92,65),(2,93,66),(3,94,34),(4,95,35),(5,96,36),(6,97,37),(7,98,38),(8,99,39),(9,67,40),(10,68,41),(11,69,42),(12,70,43),(13,71,44),(14,72,45),(15,73,46),(16,74,47),(17,75,48),(18,76,49),(19,77,50),(20,78,51),(21,79,52),(22,80,53),(23,81,54),(24,82,55),(25,83,56),(26,84,57),(27,85,58),(28,86,59),(29,87,60),(30,88,61),(31,89,62),(32,90,63),(33,91,64)], [(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)], [(1,33),(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,25),(10,24),(11,23),(12,22),(13,21),(14,20),(15,19),(16,18),(34,89),(35,88),(36,87),(37,86),(38,85),(39,84),(40,83),(41,82),(42,81),(43,80),(44,79),(45,78),(46,77),(47,76),(48,75),(49,74),(50,73),(51,72),(52,71),(53,70),(54,69),(55,68),(56,67),(57,99),(58,98),(59,97),(60,96),(61,95),(62,94),(63,93),(64,92),(65,91),(66,90)]])
C3⋊D33 is a maximal subgroup of
C3⋊S3×D11 S3×D33
C3⋊D33 is a maximal quotient of C3⋊Dic33
51 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 11A | ··· | 11E | 33A | ··· | 33AN |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 11 | ··· | 11 | 33 | ··· | 33 |
| size | 1 | 99 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
51 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | S3 | D11 | D33 |
| kernel | C3⋊D33 | C3×C33 | C33 | C32 | C3 |
| # reps | 1 | 1 | 4 | 5 | 40 |
Matrix representation of C3⋊D33 ►in GL4(𝔽67) generated by
| 58 | 41 | 0 | 0 |
| 26 | 8 | 0 | 0 |
| 0 | 0 | 58 | 41 |
| 0 | 0 | 26 | 8 |
| 31 | 57 | 0 | 0 |
| 10 | 53 | 0 | 0 |
| 0 | 0 | 64 | 52 |
| 0 | 0 | 15 | 30 |
| 31 | 57 | 0 | 0 |
| 29 | 36 | 0 | 0 |
| 0 | 0 | 36 | 10 |
| 0 | 0 | 38 | 31 |
G:=sub<GL(4,GF(67))| [58,26,0,0,41,8,0,0,0,0,58,26,0,0,41,8],[31,10,0,0,57,53,0,0,0,0,64,15,0,0,52,30],[31,29,0,0,57,36,0,0,0,0,36,38,0,0,10,31] >;
C3⋊D33 in GAP, Magma, Sage, TeX
C_3\rtimes D_{33} % in TeX
G:=Group("C3:D33"); // GroupNames label
G:=SmallGroup(198,9);
// by ID
G=gap.SmallGroup(198,9);
# by ID
G:=PCGroup([4,-2,-3,-3,-11,33,146,2883]);
// Polycyclic
G:=Group<a,b,c|a^3=b^33=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export