direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5×C3⋊C8, C3⋊C40, C15⋊5C8, C6.C20, C30.5C4, C60.6C2, C20.4S3, C12.2C10, C10.3Dic3, C4.2(C5×S3), C2.(C5×Dic3), SmallGroup(120,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C5×C3⋊C8 |
Generators and relations for C5×C3⋊C8
G = < a,b,c | a5=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C5×C3⋊C8
►Regular action on 120 points
Generators in S120
(1 15 26 51 77)(2 16 27 52 78)(3 9 28 53 79)(4 10 29 54 80)(5 11 30 55 73)(6 12 31 56 74)(7 13 32 49 75)(8 14 25 50 76)(17 36 63 85 110)(18 37 64 86 111)(19 38 57 87 112)(20 39 58 88 105)(21 40 59 81 106)(22 33 60 82 107)(23 34 61 83 108)(24 35 62 84 109)(41 103 69 94 119)(42 104 70 95 120)(43 97 71 96 113)(44 98 72 89 114)(45 99 65 90 115)(46 100 66 91 116)(47 101 67 92 117)(48 102 68 93 118)
(1 21 103)(2 104 22)(3 23 97)(4 98 24)(5 17 99)(6 100 18)(7 19 101)(8 102 20)(9 34 71)(10 72 35)(11 36 65)(12 66 37)(13 38 67)(14 68 39)(15 40 69)(16 70 33)(25 93 58)(26 59 94)(27 95 60)(28 61 96)(29 89 62)(30 63 90)(31 91 64)(32 57 92)(41 77 106)(42 107 78)(43 79 108)(44 109 80)(45 73 110)(46 111 74)(47 75 112)(48 105 76)(49 87 117)(50 118 88)(51 81 119)(52 120 82)(53 83 113)(54 114 84)(55 85 115)(56 116 86)
(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)| (1,15,26,51,77)(2,16,27,52,78)(3,9,28,53,79)(4,10,29,54,80)(5,11,30,55,73)(6,12,31,56,74)(7,13,32,49,75)(8,14,25,50,76)(17,36,63,85,110)(18,37,64,86,111)(19,38,57,87,112)(20,39,58,88,105)(21,40,59,81,106)(22,33,60,82,107)(23,34,61,83,108)(24,35,62,84,109)(41,103,69,94,119)(42,104,70,95,120)(43,97,71,96,113)(44,98,72,89,114)(45,99,65,90,115)(46,100,66,91,116)(47,101,67,92,117)(48,102,68,93,118), (1,21,103)(2,104,22)(3,23,97)(4,98,24)(5,17,99)(6,100,18)(7,19,101)(8,102,20)(9,34,71)(10,72,35)(11,36,65)(12,66,37)(13,38,67)(14,68,39)(15,40,69)(16,70,33)(25,93,58)(26,59,94)(27,95,60)(28,61,96)(29,89,62)(30,63,90)(31,91,64)(32,57,92)(41,77,106)(42,107,78)(43,79,108)(44,109,80)(45,73,110)(46,111,74)(47,75,112)(48,105,76)(49,87,117)(50,118,88)(51,81,119)(52,120,82)(53,83,113)(54,114,84)(55,85,115)(56,116,86), (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( (1,15,26,51,77)(2,16,27,52,78)(3,9,28,53,79)(4,10,29,54,80)(5,11,30,55,73)(6,12,31,56,74)(7,13,32,49,75)(8,14,25,50,76)(17,36,63,85,110)(18,37,64,86,111)(19,38,57,87,112)(20,39,58,88,105)(21,40,59,81,106)(22,33,60,82,107)(23,34,61,83,108)(24,35,62,84,109)(41,103,69,94,119)(42,104,70,95,120)(43,97,71,96,113)(44,98,72,89,114)(45,99,65,90,115)(46,100,66,91,116)(47,101,67,92,117)(48,102,68,93,118), (1,21,103)(2,104,22)(3,23,97)(4,98,24)(5,17,99)(6,100,18)(7,19,101)(8,102,20)(9,34,71)(10,72,35)(11,36,65)(12,66,37)(13,38,67)(14,68,39)(15,40,69)(16,70,33)(25,93,58)(26,59,94)(27,95,60)(28,61,96)(29,89,62)(30,63,90)(31,91,64)(32,57,92)(41,77,106)(42,107,78)(43,79,108)(44,109,80)(45,73,110)(46,111,74)(47,75,112)(48,105,76)(49,87,117)(50,118,88)(51,81,119)(52,120,82)(53,83,113)(54,114,84)(55,85,115)(56,116,86), (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([[(1,15,26,51,77),(2,16,27,52,78),(3,9,28,53,79),(4,10,29,54,80),(5,11,30,55,73),(6,12,31,56,74),(7,13,32,49,75),(8,14,25,50,76),(17,36,63,85,110),(18,37,64,86,111),(19,38,57,87,112),(20,39,58,88,105),(21,40,59,81,106),(22,33,60,82,107),(23,34,61,83,108),(24,35,62,84,109),(41,103,69,94,119),(42,104,70,95,120),(43,97,71,96,113),(44,98,72,89,114),(45,99,65,90,115),(46,100,66,91,116),(47,101,67,92,117),(48,102,68,93,118)], [(1,21,103),(2,104,22),(3,23,97),(4,98,24),(5,17,99),(6,100,18),(7,19,101),(8,102,20),(9,34,71),(10,72,35),(11,36,65),(12,66,37),(13,38,67),(14,68,39),(15,40,69),(16,70,33),(25,93,58),(26,59,94),(27,95,60),(28,61,96),(29,89,62),(30,63,90),(31,91,64),(32,57,92),(41,77,106),(42,107,78),(43,79,108),(44,109,80),(45,73,110),(46,111,74),(47,75,112),(48,105,76),(49,87,117),(50,118,88),(51,81,119),(52,120,82),(53,83,113),(54,114,84),(55,85,115),(56,116,86)], [(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)]])
C5×C3⋊C8 is a maximal subgroup of
D15⋊2C8 C20.32D6 D30.5C4 C3⋊D40 C6.D20 C15⋊SD16 C3⋊Dic20 S3×C40
60 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 5A | 5B | 5C | 5D | 6 | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 12A | 12B | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | 30B | 30C | 30D | 40A | ··· | 40P | 60A | ··· | 60H |
| order | 1 | 2 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||
| image | C1 | C2 | C4 | C5 | C8 | C10 | C20 | C40 | S3 | Dic3 | C3⋊C8 | C5×S3 | C5×Dic3 | C5×C3⋊C8 |
| kernel | C5×C3⋊C8 | C60 | C30 | C3⋊C8 | C15 | C12 | C6 | C3 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 4 | 8 | 16 | 1 | 1 | 2 | 4 | 4 | 8 |
Matrix representation of C5×C3⋊C8 ►in GL2(𝔽41) generated by
| 16 | 0 |
| 0 | 16 |
| 0 | 37 |
| 31 | 40 |
| 3 | 29 |
| 0 | 38 |
G:=sub<GL(2,GF(41))| [16,0,0,16],[0,31,37,40],[3,0,29,38] >;
C5×C3⋊C8 in GAP, Magma, Sage, TeX
C_5\times C_3\rtimes C_8
% in TeX
G:=Group("C5xC3:C8"); // GroupNames label
G:=SmallGroup(120,1);
// by ID
G=gap.SmallGroup(120,1);
# by ID
G:=PCGroup([5,-2,-5,-2,-2,-3,50,42,2004]);
// Polycyclic
G:=Group<a,b,c|a^5=b^3=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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