direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C3×D5⋊C8, D5⋊C24, C60.6C4, C12.6F5, C20.3C12, D10.2C12, C5⋊C8⋊3C6, C15⋊4(C2×C8), C5⋊1(C2×C24), (C3×D5)⋊3C8, C4.3(C3×F5), C2.1(C6×F5), (C4×D5).5C6, (C6×D5).6C4, C6.15(C2×F5), C30.15(C2×C4), C10.1(C2×C12), (D5×C12).12C2, Dic5.5(C2×C6), (C3×Dic5).25C22, (C3×C5⋊C8)⋊7C2, SmallGroup(240,111)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C3×D5⋊C8 |
Generators and relations for C3×D5⋊C8
G = < a,b,c,d | a3=b5=c2=d8=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b2c >
Smallest permutation representation of C3×D5⋊C8
►On 120 points
Generators in S120
(1 100 60)(2 101 61)(3 102 62)(4 103 63)(5 104 64)(6 97 57)(7 98 58)(8 99 59)(9 86 46)(10 87 47)(11 88 48)(12 81 41)(13 82 42)(14 83 43)(15 84 44)(16 85 45)(17 95 120)(18 96 113)(19 89 114)(20 90 115)(21 91 116)(22 92 117)(23 93 118)(24 94 119)(25 110 66)(26 111 67)(27 112 68)(28 105 69)(29 106 70)(30 107 71)(31 108 72)(32 109 65)(33 56 74)(34 49 75)(35 50 76)(36 51 77)(37 52 78)(38 53 79)(39 54 80)(40 55 73)
(1 105 22 46 49)(2 47 106 50 23)(3 51 48 24 107)(4 17 52 108 41)(5 109 18 42 53)(6 43 110 54 19)(7 55 44 20 111)(8 21 56 112 45)(9 75 100 69 92)(10 70 76 93 101)(11 94 71 102 77)(12 103 95 78 72)(13 79 104 65 96)(14 66 80 89 97)(15 90 67 98 73)(16 99 91 74 68)(25 39 114 57 83)(26 58 40 84 115)(27 85 59 116 33)(28 117 86 34 60)(29 35 118 61 87)(30 62 36 88 119)(31 81 63 120 37)(32 113 82 38 64)
(1 49)(2 23)(3 107)(4 41)(5 53)(6 19)(7 111)(8 45)(9 69)(10 76)(12 103)(13 65)(14 80)(16 99)(17 108)(20 55)(21 112)(24 51)(26 58)(27 116)(28 86)(30 62)(31 120)(32 82)(34 60)(35 87)(36 119)(38 64)(39 83)(40 115)(42 109)(43 54)(46 105)(47 50)(57 114)(59 85)(61 118)(63 81)(67 98)(68 91)(71 102)(72 95)(73 90)(75 100)(77 94)(79 104)(89 97)(93 101)
(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)| (1,100,60)(2,101,61)(3,102,62)(4,103,63)(5,104,64)(6,97,57)(7,98,58)(8,99,59)(9,86,46)(10,87,47)(11,88,48)(12,81,41)(13,82,42)(14,83,43)(15,84,44)(16,85,45)(17,95,120)(18,96,113)(19,89,114)(20,90,115)(21,91,116)(22,92,117)(23,93,118)(24,94,119)(25,110,66)(26,111,67)(27,112,68)(28,105,69)(29,106,70)(30,107,71)(31,108,72)(32,109,65)(33,56,74)(34,49,75)(35,50,76)(36,51,77)(37,52,78)(38,53,79)(39,54,80)(40,55,73), (1,105,22,46,49)(2,47,106,50,23)(3,51,48,24,107)(4,17,52,108,41)(5,109,18,42,53)(6,43,110,54,19)(7,55,44,20,111)(8,21,56,112,45)(9,75,100,69,92)(10,70,76,93,101)(11,94,71,102,77)(12,103,95,78,72)(13,79,104,65,96)(14,66,80,89,97)(15,90,67,98,73)(16,99,91,74,68)(25,39,114,57,83)(26,58,40,84,115)(27,85,59,116,33)(28,117,86,34,60)(29,35,118,61,87)(30,62,36,88,119)(31,81,63,120,37)(32,113,82,38,64), (1,49)(2,23)(3,107)(4,41)(5,53)(6,19)(7,111)(8,45)(9,69)(10,76)(12,103)(13,65)(14,80)(16,99)(17,108)(20,55)(21,112)(24,51)(26,58)(27,116)(28,86)(30,62)(31,120)(32,82)(34,60)(35,87)(36,119)(38,64)(39,83)(40,115)(42,109)(43,54)(46,105)(47,50)(57,114)(59,85)(61,118)(63,81)(67,98)(68,91)(71,102)(72,95)(73,90)(75,100)(77,94)(79,104)(89,97)(93,101), (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( (1,100,60)(2,101,61)(3,102,62)(4,103,63)(5,104,64)(6,97,57)(7,98,58)(8,99,59)(9,86,46)(10,87,47)(11,88,48)(12,81,41)(13,82,42)(14,83,43)(15,84,44)(16,85,45)(17,95,120)(18,96,113)(19,89,114)(20,90,115)(21,91,116)(22,92,117)(23,93,118)(24,94,119)(25,110,66)(26,111,67)(27,112,68)(28,105,69)(29,106,70)(30,107,71)(31,108,72)(32,109,65)(33,56,74)(34,49,75)(35,50,76)(36,51,77)(37,52,78)(38,53,79)(39,54,80)(40,55,73), (1,105,22,46,49)(2,47,106,50,23)(3,51,48,24,107)(4,17,52,108,41)(5,109,18,42,53)(6,43,110,54,19)(7,55,44,20,111)(8,21,56,112,45)(9,75,100,69,92)(10,70,76,93,101)(11,94,71,102,77)(12,103,95,78,72)(13,79,104,65,96)(14,66,80,89,97)(15,90,67,98,73)(16,99,91,74,68)(25,39,114,57,83)(26,58,40,84,115)(27,85,59,116,33)(28,117,86,34,60)(29,35,118,61,87)(30,62,36,88,119)(31,81,63,120,37)(32,113,82,38,64), (1,49)(2,23)(3,107)(4,41)(5,53)(6,19)(7,111)(8,45)(9,69)(10,76)(12,103)(13,65)(14,80)(16,99)(17,108)(20,55)(21,112)(24,51)(26,58)(27,116)(28,86)(30,62)(31,120)(32,82)(34,60)(35,87)(36,119)(38,64)(39,83)(40,115)(42,109)(43,54)(46,105)(47,50)(57,114)(59,85)(61,118)(63,81)(67,98)(68,91)(71,102)(72,95)(73,90)(75,100)(77,94)(79,104)(89,97)(93,101), (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([[(1,100,60),(2,101,61),(3,102,62),(4,103,63),(5,104,64),(6,97,57),(7,98,58),(8,99,59),(9,86,46),(10,87,47),(11,88,48),(12,81,41),(13,82,42),(14,83,43),(15,84,44),(16,85,45),(17,95,120),(18,96,113),(19,89,114),(20,90,115),(21,91,116),(22,92,117),(23,93,118),(24,94,119),(25,110,66),(26,111,67),(27,112,68),(28,105,69),(29,106,70),(30,107,71),(31,108,72),(32,109,65),(33,56,74),(34,49,75),(35,50,76),(36,51,77),(37,52,78),(38,53,79),(39,54,80),(40,55,73)], [(1,105,22,46,49),(2,47,106,50,23),(3,51,48,24,107),(4,17,52,108,41),(5,109,18,42,53),(6,43,110,54,19),(7,55,44,20,111),(8,21,56,112,45),(9,75,100,69,92),(10,70,76,93,101),(11,94,71,102,77),(12,103,95,78,72),(13,79,104,65,96),(14,66,80,89,97),(15,90,67,98,73),(16,99,91,74,68),(25,39,114,57,83),(26,58,40,84,115),(27,85,59,116,33),(28,117,86,34,60),(29,35,118,61,87),(30,62,36,88,119),(31,81,63,120,37),(32,113,82,38,64)], [(1,49),(2,23),(3,107),(4,41),(5,53),(6,19),(7,111),(8,45),(9,69),(10,76),(12,103),(13,65),(14,80),(16,99),(17,108),(20,55),(21,112),(24,51),(26,58),(27,116),(28,86),(30,62),(31,120),(32,82),(34,60),(35,87),(36,119),(38,64),(39,83),(40,115),(42,109),(43,54),(46,105),(47,50),(57,114),(59,85),(61,118),(63,81),(67,98),(68,91),(71,102),(72,95),(73,90),(75,100),(77,94),(79,104),(89,97),(93,101)], [(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×D5⋊C8 is a maximal subgroup of
C30.C42 C30.4C42 D12⋊F5 Dic30⋊C4 F5×C24 D12.2F5 D60.C4 C5⋊C8⋊D6
C3×D5⋊C8 is a maximal quotient of
C12×C5⋊C8
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 5 | 6A | 6B | 6C | 6D | 6E | 6F | 8A | ··· | 8H | 10 | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 20A | 20B | 24A | ··· | 24P | 30A | 30B | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 24 | ··· | 24 | 30 | 30 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 1 | 1 | 5 | 5 | 4 | 1 | 1 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 4 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 4 | 4 | 4 | 4 | 5 | ··· | 5 | 4 | 4 | 4 | 4 | 4 | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C12 | C12 | C24 | F5 | C2×F5 | C3×F5 | D5⋊C8 | C6×F5 | C3×D5⋊C8 |
| kernel | C3×D5⋊C8 | C3×C5⋊C8 | D5×C12 | D5⋊C8 | C60 | C6×D5 | C5⋊C8 | C4×D5 | C3×D5 | C20 | D10 | D5 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 8 | 4 | 4 | 16 | 1 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of C3×D5⋊C8 ►in GL4(𝔽241) generated by
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 240 | 1 | 0 | 0 |
| 240 | 0 | 1 | 0 |
| 240 | 0 | 0 | 1 |
| 240 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 |
| 240 | 0 | 0 | 1 |
| 240 | 0 | 1 | 0 |
| 240 | 1 | 0 | 0 |
| 179 | 62 | 38 | 0 |
| 217 | 62 | 0 | 179 |
| 179 | 0 | 62 | 217 |
| 0 | 38 | 62 | 179 |
G:=sub<GL(4,GF(241))| [15,0,0,0,0,15,0,0,0,0,15,0,0,0,0,15],[240,240,240,240,1,0,0,0,0,1,0,0,0,0,1,0],[240,240,240,240,0,0,0,1,0,0,1,0,0,1,0,0],[179,217,179,0,62,62,0,38,38,0,62,62,0,179,217,179] >;
C3×D5⋊C8 in GAP, Magma, Sage, TeX
C_3\times D_5\rtimes C_8
% in TeX
G:=Group("C3xD5:C8"); // GroupNames label
G:=SmallGroup(240,111);
// by ID
G=gap.SmallGroup(240,111);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-5,72,151,69,3461,599]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^5=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^2*c>;
// generators/relations
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