metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊4S3, C24⋊5D5, C8⋊3D15, C120⋊5C2, D30.3C4, C20.48D6, C4.13D30, C12.49D10, C15⋊10M4(2), C60.55C22, Dic15.3C4, C15⋊3C8⋊4C2, C6.6(C4×D5), C5⋊4(C8⋊S3), C3⋊2(C8⋊D5), C2.3(C4×D15), C10.13(C4×S3), C30.36(C2×C4), (C4×D15).2C2, SmallGroup(240,66)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊S3
G = < a,b,c | a40=b3=c2=1, ab=ba, cac=a29, cbc=b-1 >
Smallest permutation representation of C40⋊S3
►On 120 points
Generators in S120
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)
(1 99 78)(2 100 79)(3 101 80)(4 102 41)(5 103 42)(6 104 43)(7 105 44)(8 106 45)(9 107 46)(10 108 47)(11 109 48)(12 110 49)(13 111 50)(14 112 51)(15 113 52)(16 114 53)(17 115 54)(18 116 55)(19 117 56)(20 118 57)(21 119 58)(22 120 59)(23 81 60)(24 82 61)(25 83 62)(26 84 63)(27 85 64)(28 86 65)(29 87 66)(30 88 67)(31 89 68)(32 90 69)(33 91 70)(34 92 71)(35 93 72)(36 94 73)(37 95 74)(38 96 75)(39 97 76)(40 98 77)
(2 30)(3 19)(4 8)(5 37)(6 26)(7 15)(9 33)(10 22)(12 40)(13 29)(14 18)(16 36)(17 25)(20 32)(23 39)(24 28)(27 35)(34 38)(41 106)(42 95)(43 84)(44 113)(45 102)(46 91)(47 120)(48 109)(49 98)(50 87)(51 116)(52 105)(53 94)(54 83)(55 112)(56 101)(57 90)(58 119)(59 108)(60 97)(61 86)(62 115)(63 104)(64 93)(65 82)(66 111)(67 100)(68 89)(69 118)(70 107)(71 96)(72 85)(73 114)(74 103)(75 92)(76 81)(77 110)(78 99)(79 88)(80 117)
G:=sub<Sym(120)| (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)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,99,78)(2,100,79)(3,101,80)(4,102,41)(5,103,42)(6,104,43)(7,105,44)(8,106,45)(9,107,46)(10,108,47)(11,109,48)(12,110,49)(13,111,50)(14,112,51)(15,113,52)(16,114,53)(17,115,54)(18,116,55)(19,117,56)(20,118,57)(21,119,58)(22,120,59)(23,81,60)(24,82,61)(25,83,62)(26,84,63)(27,85,64)(28,86,65)(29,87,66)(30,88,67)(31,89,68)(32,90,69)(33,91,70)(34,92,71)(35,93,72)(36,94,73)(37,95,74)(38,96,75)(39,97,76)(40,98,77), (2,30)(3,19)(4,8)(5,37)(6,26)(7,15)(9,33)(10,22)(12,40)(13,29)(14,18)(16,36)(17,25)(20,32)(23,39)(24,28)(27,35)(34,38)(41,106)(42,95)(43,84)(44,113)(45,102)(46,91)(47,120)(48,109)(49,98)(50,87)(51,116)(52,105)(53,94)(54,83)(55,112)(56,101)(57,90)(58,119)(59,108)(60,97)(61,86)(62,115)(63,104)(64,93)(65,82)(66,111)(67,100)(68,89)(69,118)(70,107)(71,96)(72,85)(73,114)(74,103)(75,92)(76,81)(77,110)(78,99)(79,88)(80,117)>;
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)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,99,78)(2,100,79)(3,101,80)(4,102,41)(5,103,42)(6,104,43)(7,105,44)(8,106,45)(9,107,46)(10,108,47)(11,109,48)(12,110,49)(13,111,50)(14,112,51)(15,113,52)(16,114,53)(17,115,54)(18,116,55)(19,117,56)(20,118,57)(21,119,58)(22,120,59)(23,81,60)(24,82,61)(25,83,62)(26,84,63)(27,85,64)(28,86,65)(29,87,66)(30,88,67)(31,89,68)(32,90,69)(33,91,70)(34,92,71)(35,93,72)(36,94,73)(37,95,74)(38,96,75)(39,97,76)(40,98,77), (2,30)(3,19)(4,8)(5,37)(6,26)(7,15)(9,33)(10,22)(12,40)(13,29)(14,18)(16,36)(17,25)(20,32)(23,39)(24,28)(27,35)(34,38)(41,106)(42,95)(43,84)(44,113)(45,102)(46,91)(47,120)(48,109)(49,98)(50,87)(51,116)(52,105)(53,94)(54,83)(55,112)(56,101)(57,90)(58,119)(59,108)(60,97)(61,86)(62,115)(63,104)(64,93)(65,82)(66,111)(67,100)(68,89)(69,118)(70,107)(71,96)(72,85)(73,114)(74,103)(75,92)(76,81)(77,110)(78,99)(79,88)(80,117) );
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),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)], [(1,99,78),(2,100,79),(3,101,80),(4,102,41),(5,103,42),(6,104,43),(7,105,44),(8,106,45),(9,107,46),(10,108,47),(11,109,48),(12,110,49),(13,111,50),(14,112,51),(15,113,52),(16,114,53),(17,115,54),(18,116,55),(19,117,56),(20,118,57),(21,119,58),(22,120,59),(23,81,60),(24,82,61),(25,83,62),(26,84,63),(27,85,64),(28,86,65),(29,87,66),(30,88,67),(31,89,68),(32,90,69),(33,91,70),(34,92,71),(35,93,72),(36,94,73),(37,95,74),(38,96,75),(39,97,76),(40,98,77)], [(2,30),(3,19),(4,8),(5,37),(6,26),(7,15),(9,33),(10,22),(12,40),(13,29),(14,18),(16,36),(17,25),(20,32),(23,39),(24,28),(27,35),(34,38),(41,106),(42,95),(43,84),(44,113),(45,102),(46,91),(47,120),(48,109),(49,98),(50,87),(51,116),(52,105),(53,94),(54,83),(55,112),(56,101),(57,90),(58,119),(59,108),(60,97),(61,86),(62,115),(63,104),(64,93),(65,82),(66,111),(67,100),(68,89),(69,118),(70,107),(71,96),(72,85),(73,114),(74,103),(75,92),(76,81),(77,110),(78,99),(79,88),(80,117)]])
C40⋊S3 is a maximal subgroup of
D5×C8⋊S3 S3×C8⋊D5 D24⋊6D5 C40⋊8D6 C40.34D6 C40.55D6 D30.3D4 D30.4D4 D60.6C4 M4(2)×D15 D60.3C4 D8⋊D15 Q8⋊3D30 SD16⋊D15 Q16⋊D15
C40⋊S3 is a maximal quotient of
C60.26Q8 C120⋊13C4 D30⋊3C8
66 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 5A | 5B | 6 | 8A | 8B | 8C | 8D | 10A | 10B | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 40A | ··· | 40H | 60A | ··· | 60H | 120A | ··· | 120P |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 30 | 2 | 1 | 1 | 30 | 2 | 2 | 2 | 2 | 2 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D5 | D6 | M4(2) | D10 | C4×S3 | D15 | C4×D5 | C8⋊S3 | D30 | C8⋊D5 | C4×D15 | C40⋊S3 |
| kernel | C40⋊S3 | C15⋊3C8 | C120 | C4×D15 | Dic15 | D30 | C40 | C24 | C20 | C15 | C12 | C10 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of C40⋊S3 ►in GL2(𝔽29) generated by
| 3 | 9 |
| 8 | 20 |
| 20 | 9 |
| 8 | 8 |
| 7 | 2 |
| 5 | 22 |
G:=sub<GL(2,GF(29))| [3,8,9,20],[20,8,9,8],[7,5,2,22] >;
C40⋊S3 in GAP, Magma, Sage, TeX
C_{40}\rtimes S_3 % in TeX
G:=Group("C40:S3"); // GroupNames label
G:=SmallGroup(240,66);
// by ID
G=gap.SmallGroup(240,66);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,121,31,50,964,6917]);
// Polycyclic
G:=Group<a,b,c|a^40=b^3=c^2=1,a*b=b*a,c*a*c=a^29,c*b*c=b^-1>;
// generators/relations
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