direct product, metabelian, supersoluble, monomial, A-group
Aliases: C11×C3⋊S3, C33⋊3S3, C32⋊2C22, C3⋊(S3×C11), (C3×C33)⋊5C2, SmallGroup(198,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C11×C3⋊S3 |
Generators and relations for C11×C3⋊S3
G = < a,b,c,d | a11=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Smallest permutation representation of C11×C3⋊S3
►On 99 points
Generators in S99
(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 78 27)(2 79 28)(3 80 29)(4 81 30)(5 82 31)(6 83 32)(7 84 33)(8 85 23)(9 86 24)(10 87 25)(11 88 26)(12 96 36)(13 97 37)(14 98 38)(15 99 39)(16 89 40)(17 90 41)(18 91 42)(19 92 43)(20 93 44)(21 94 34)(22 95 35)(45 66 74)(46 56 75)(47 57 76)(48 58 77)(49 59 67)(50 60 68)(51 61 69)(52 62 70)(53 63 71)(54 64 72)(55 65 73)
(1 52 17)(2 53 18)(3 54 19)(4 55 20)(5 45 21)(6 46 22)(7 47 12)(8 48 13)(9 49 14)(10 50 15)(11 51 16)(23 77 37)(24 67 38)(25 68 39)(26 69 40)(27 70 41)(28 71 42)(29 72 43)(30 73 44)(31 74 34)(32 75 35)(33 76 36)(56 95 83)(57 96 84)(58 97 85)(59 98 86)(60 99 87)(61 89 88)(62 90 78)(63 91 79)(64 92 80)(65 93 81)(66 94 82)
(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 45)(22 46)(23 85)(24 86)(25 87)(26 88)(27 78)(28 79)(29 80)(30 81)(31 82)(32 83)(33 84)(34 66)(35 56)(36 57)(37 58)(38 59)(39 60)(40 61)(41 62)(42 63)(43 64)(44 65)(67 98)(68 99)(69 89)(70 90)(71 91)(72 92)(73 93)(74 94)(75 95)(76 96)(77 97)
G:=sub<Sym(99)| (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), (1,78,27)(2,79,28)(3,80,29)(4,81,30)(5,82,31)(6,83,32)(7,84,33)(8,85,23)(9,86,24)(10,87,25)(11,88,26)(12,96,36)(13,97,37)(14,98,38)(15,99,39)(16,89,40)(17,90,41)(18,91,42)(19,92,43)(20,93,44)(21,94,34)(22,95,35)(45,66,74)(46,56,75)(47,57,76)(48,58,77)(49,59,67)(50,60,68)(51,61,69)(52,62,70)(53,63,71)(54,64,72)(55,65,73), (1,52,17)(2,53,18)(3,54,19)(4,55,20)(5,45,21)(6,46,22)(7,47,12)(8,48,13)(9,49,14)(10,50,15)(11,51,16)(23,77,37)(24,67,38)(25,68,39)(26,69,40)(27,70,41)(28,71,42)(29,72,43)(30,73,44)(31,74,34)(32,75,35)(33,76,36)(56,95,83)(57,96,84)(58,97,85)(59,98,86)(60,99,87)(61,89,88)(62,90,78)(63,91,79)(64,92,80)(65,93,81)(66,94,82), (12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,45)(22,46)(23,85)(24,86)(25,87)(26,88)(27,78)(28,79)(29,80)(30,81)(31,82)(32,83)(33,84)(34,66)(35,56)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63)(43,64)(44,65)(67,98)(68,99)(69,89)(70,90)(71,91)(72,92)(73,93)(74,94)(75,95)(76,96)(77,97)>;
G:=Group( (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), (1,78,27)(2,79,28)(3,80,29)(4,81,30)(5,82,31)(6,83,32)(7,84,33)(8,85,23)(9,86,24)(10,87,25)(11,88,26)(12,96,36)(13,97,37)(14,98,38)(15,99,39)(16,89,40)(17,90,41)(18,91,42)(19,92,43)(20,93,44)(21,94,34)(22,95,35)(45,66,74)(46,56,75)(47,57,76)(48,58,77)(49,59,67)(50,60,68)(51,61,69)(52,62,70)(53,63,71)(54,64,72)(55,65,73), (1,52,17)(2,53,18)(3,54,19)(4,55,20)(5,45,21)(6,46,22)(7,47,12)(8,48,13)(9,49,14)(10,50,15)(11,51,16)(23,77,37)(24,67,38)(25,68,39)(26,69,40)(27,70,41)(28,71,42)(29,72,43)(30,73,44)(31,74,34)(32,75,35)(33,76,36)(56,95,83)(57,96,84)(58,97,85)(59,98,86)(60,99,87)(61,89,88)(62,90,78)(63,91,79)(64,92,80)(65,93,81)(66,94,82), (12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,45)(22,46)(23,85)(24,86)(25,87)(26,88)(27,78)(28,79)(29,80)(30,81)(31,82)(32,83)(33,84)(34,66)(35,56)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63)(43,64)(44,65)(67,98)(68,99)(69,89)(70,90)(71,91)(72,92)(73,93)(74,94)(75,95)(76,96)(77,97) );
G=PermutationGroup([[(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)], [(1,78,27),(2,79,28),(3,80,29),(4,81,30),(5,82,31),(6,83,32),(7,84,33),(8,85,23),(9,86,24),(10,87,25),(11,88,26),(12,96,36),(13,97,37),(14,98,38),(15,99,39),(16,89,40),(17,90,41),(18,91,42),(19,92,43),(20,93,44),(21,94,34),(22,95,35),(45,66,74),(46,56,75),(47,57,76),(48,58,77),(49,59,67),(50,60,68),(51,61,69),(52,62,70),(53,63,71),(54,64,72),(55,65,73)], [(1,52,17),(2,53,18),(3,54,19),(4,55,20),(5,45,21),(6,46,22),(7,47,12),(8,48,13),(9,49,14),(10,50,15),(11,51,16),(23,77,37),(24,67,38),(25,68,39),(26,69,40),(27,70,41),(28,71,42),(29,72,43),(30,73,44),(31,74,34),(32,75,35),(33,76,36),(56,95,83),(57,96,84),(58,97,85),(59,98,86),(60,99,87),(61,89,88),(62,90,78),(63,91,79),(64,92,80),(65,93,81),(66,94,82)], [(12,47),(13,48),(14,49),(15,50),(16,51),(17,52),(18,53),(19,54),(20,55),(21,45),(22,46),(23,85),(24,86),(25,87),(26,88),(27,78),(28,79),(29,80),(30,81),(31,82),(32,83),(33,84),(34,66),(35,56),(36,57),(37,58),(38,59),(39,60),(40,61),(41,62),(42,63),(43,64),(44,65),(67,98),(68,99),(69,89),(70,90),(71,91),(72,92),(73,93),(74,94),(75,95),(76,96),(77,97)]])
C11×C3⋊S3 is a maximal subgroup of
C32⋊Dic11 S32×C11 D33⋊S3
66 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 11A | ··· | 11J | 22A | ··· | 22J | 33A | ··· | 33AN |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 11 | ··· | 11 | 22 | ··· | 22 | 33 | ··· | 33 |
| size | 1 | 9 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 9 | ··· | 9 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C11 | C22 | S3 | S3×C11 |
| kernel | C11×C3⋊S3 | C3×C33 | C3⋊S3 | C32 | C33 | C3 |
| # reps | 1 | 1 | 10 | 10 | 4 | 40 |
Matrix representation of C11×C3⋊S3 ►in GL4(𝔽67) generated by
| 62 | 0 | 0 | 0 |
| 0 | 62 | 0 | 0 |
| 0 | 0 | 62 | 0 |
| 0 | 0 | 0 | 62 |
| 66 | 1 | 0 | 0 |
| 66 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 66 | 1 | 0 | 0 |
| 66 | 0 | 0 | 0 |
| 0 | 0 | 66 | 66 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 66 | 66 |
G:=sub<GL(4,GF(67))| [62,0,0,0,0,62,0,0,0,0,62,0,0,0,0,62],[66,66,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[66,66,0,0,1,0,0,0,0,0,66,1,0,0,66,0],[0,1,0,0,1,0,0,0,0,0,1,66,0,0,0,66] >;
C11×C3⋊S3 in GAP, Magma, Sage, TeX
C_{11}\times C_3\rtimes S_3 % in TeX
G:=Group("C11xC3:S3"); // GroupNames label
G:=SmallGroup(198,8);
// by ID
G=gap.SmallGroup(198,8);
# by ID
G:=PCGroup([4,-2,-11,-3,-3,530,2115]);
// Polycyclic
G:=Group<a,b,c,d|a^11=b^3=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
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