direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×C5⋊2C8, C15⋊4C8, C5⋊2C24, C20.2C6, C60.5C2, C30.4C4, C12.4D5, C10.2C12, C6.2Dic5, C4.2(C3×D5), C2.(C3×Dic5), SmallGroup(120,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C3×C5⋊2C8 |
Generators and relations for C3×C5⋊2C8
G = < a,b,c | a3=b5=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×C5⋊2C8
►Regular action on 120 points
Generators in S120
(1 100 65)(2 101 66)(3 102 67)(4 103 68)(5 104 69)(6 97 70)(7 98 71)(8 99 72)(9 86 43)(10 87 44)(11 88 45)(12 81 46)(13 82 47)(14 83 48)(15 84 41)(16 85 42)(17 55 58)(18 56 59)(19 49 60)(20 50 61)(21 51 62)(22 52 63)(23 53 64)(24 54 57)(25 90 115)(26 91 116)(27 92 117)(28 93 118)(29 94 119)(30 95 120)(31 96 113)(32 89 114)(33 111 74)(34 112 75)(35 105 76)(36 106 77)(37 107 78)(38 108 79)(39 109 80)(40 110 73)
(1 48 27 51 112)(2 105 52 28 41)(3 42 29 53 106)(4 107 54 30 43)(5 44 31 55 108)(6 109 56 32 45)(7 46 25 49 110)(8 111 50 26 47)(9 103 78 57 95)(10 96 58 79 104)(11 97 80 59 89)(12 90 60 73 98)(13 99 74 61 91)(14 92 62 75 100)(15 101 76 63 93)(16 94 64 77 102)(17 38 69 87 113)(18 114 88 70 39)(19 40 71 81 115)(20 116 82 72 33)(21 34 65 83 117)(22 118 84 66 35)(23 36 67 85 119)(24 120 86 68 37)
(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)
G:=sub<Sym(120)| (1,100,65)(2,101,66)(3,102,67)(4,103,68)(5,104,69)(6,97,70)(7,98,71)(8,99,72)(9,86,43)(10,87,44)(11,88,45)(12,81,46)(13,82,47)(14,83,48)(15,84,41)(16,85,42)(17,55,58)(18,56,59)(19,49,60)(20,50,61)(21,51,62)(22,52,63)(23,53,64)(24,54,57)(25,90,115)(26,91,116)(27,92,117)(28,93,118)(29,94,119)(30,95,120)(31,96,113)(32,89,114)(33,111,74)(34,112,75)(35,105,76)(36,106,77)(37,107,78)(38,108,79)(39,109,80)(40,110,73), (1,48,27,51,112)(2,105,52,28,41)(3,42,29,53,106)(4,107,54,30,43)(5,44,31,55,108)(6,109,56,32,45)(7,46,25,49,110)(8,111,50,26,47)(9,103,78,57,95)(10,96,58,79,104)(11,97,80,59,89)(12,90,60,73,98)(13,99,74,61,91)(14,92,62,75,100)(15,101,76,63,93)(16,94,64,77,102)(17,38,69,87,113)(18,114,88,70,39)(19,40,71,81,115)(20,116,82,72,33)(21,34,65,83,117)(22,118,84,66,35)(23,36,67,85,119)(24,120,86,68,37), (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)>;
G:=Group( (1,100,65)(2,101,66)(3,102,67)(4,103,68)(5,104,69)(6,97,70)(7,98,71)(8,99,72)(9,86,43)(10,87,44)(11,88,45)(12,81,46)(13,82,47)(14,83,48)(15,84,41)(16,85,42)(17,55,58)(18,56,59)(19,49,60)(20,50,61)(21,51,62)(22,52,63)(23,53,64)(24,54,57)(25,90,115)(26,91,116)(27,92,117)(28,93,118)(29,94,119)(30,95,120)(31,96,113)(32,89,114)(33,111,74)(34,112,75)(35,105,76)(36,106,77)(37,107,78)(38,108,79)(39,109,80)(40,110,73), (1,48,27,51,112)(2,105,52,28,41)(3,42,29,53,106)(4,107,54,30,43)(5,44,31,55,108)(6,109,56,32,45)(7,46,25,49,110)(8,111,50,26,47)(9,103,78,57,95)(10,96,58,79,104)(11,97,80,59,89)(12,90,60,73,98)(13,99,74,61,91)(14,92,62,75,100)(15,101,76,63,93)(16,94,64,77,102)(17,38,69,87,113)(18,114,88,70,39)(19,40,71,81,115)(20,116,82,72,33)(21,34,65,83,117)(22,118,84,66,35)(23,36,67,85,119)(24,120,86,68,37), (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) );
G=PermutationGroup([[(1,100,65),(2,101,66),(3,102,67),(4,103,68),(5,104,69),(6,97,70),(7,98,71),(8,99,72),(9,86,43),(10,87,44),(11,88,45),(12,81,46),(13,82,47),(14,83,48),(15,84,41),(16,85,42),(17,55,58),(18,56,59),(19,49,60),(20,50,61),(21,51,62),(22,52,63),(23,53,64),(24,54,57),(25,90,115),(26,91,116),(27,92,117),(28,93,118),(29,94,119),(30,95,120),(31,96,113),(32,89,114),(33,111,74),(34,112,75),(35,105,76),(36,106,77),(37,107,78),(38,108,79),(39,109,80),(40,110,73)], [(1,48,27,51,112),(2,105,52,28,41),(3,42,29,53,106),(4,107,54,30,43),(5,44,31,55,108),(6,109,56,32,45),(7,46,25,49,110),(8,111,50,26,47),(9,103,78,57,95),(10,96,58,79,104),(11,97,80,59,89),(12,90,60,73,98),(13,99,74,61,91),(14,92,62,75,100),(15,101,76,63,93),(16,94,64,77,102),(17,38,69,87,113),(18,114,88,70,39),(19,40,71,81,115),(20,116,82,72,33),(21,34,65,83,117),(22,118,84,66,35),(23,36,67,85,119),(24,120,86,68,37)], [(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)]])
C3×C5⋊2C8 is a maximal subgroup of
C15⋊C16 D15⋊2C8 D6.Dic5 D30.5C4 C5⋊D24 D12.D5 Dic6⋊D5 C5⋊Dic12 D5×C24 SL2(𝔽3).Dic5
48 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5A | 5B | 6A | 6B | 8A | 8B | 8C | 8D | 10A | 10B | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 24A | ··· | 24H | 30A | 30B | 30C | 30D | 60A | ··· | 60H |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 30 | 30 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 5 | ··· | 5 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C24 | D5 | Dic5 | C3×D5 | C5⋊2C8 | C3×Dic5 | C3×C5⋊2C8 |
| kernel | C3×C5⋊2C8 | C60 | C5⋊2C8 | C30 | C20 | C15 | C10 | C5 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C3×C5⋊2C8 ►in GL3(𝔽241) generated by
| 225 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 51 | 240 |
| 0 | 1 | 0 |
| 240 | 0 | 0 |
| 0 | 102 | 231 |
| 0 | 131 | 139 |
G:=sub<GL(3,GF(241))| [225,0,0,0,1,0,0,0,1],[1,0,0,0,51,1,0,240,0],[240,0,0,0,102,131,0,231,139] >;
C3×C5⋊2C8 in GAP, Magma, Sage, TeX
C_3\times C_5\rtimes_2C_8
% in TeX
G:=Group("C3xC5:2C8"); // GroupNames label
G:=SmallGroup(120,2);
// by ID
G=gap.SmallGroup(120,2);
# by ID
G:=PCGroup([5,-2,-3,-2,-2,-5,30,42,2404]);
// Polycyclic
G:=Group<a,b,c|a^3=b^5=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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