direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C4×D15, C20⋊8D6, C12⋊8D10, C60⋊9C22, C22.9D30, C30.29C23, D30.15C22, Dic15⋊10C22, C6⋊2(C4×D5), C10⋊3(C4×S3), C30⋊7(C2×C4), (C2×C60)⋊7C2, (C2×C20)⋊5S3, (C2×C12)⋊5D5, C15⋊8(C22×C4), (C2×C6).27D10, (C2×C10).27D6, C2.1(C22×D15), C6.29(C22×D5), (C2×Dic15)⋊11C2, (C2×C30).28C22, C10.29(C22×S3), (C22×D15).4C2, C5⋊4(S3×C2×C4), C3⋊3(C2×C4×D5), SmallGroup(240,176)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C2×C4×D15 |
Generators and relations for C2×C4×D15
G = < a,b,c,d | a2=b4=c15=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 488 in 108 conjugacy classes, 51 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C22×C4, Dic5, C20, D10, C2×C10, C4×S3, C2×Dic3, C2×C12, C22×S3, D15, C30, C30, C4×D5, C2×Dic5, C2×C20, C22×D5, S3×C2×C4, Dic15, C60, D30, C2×C30, C2×C4×D5, C4×D15, C2×Dic15, C2×C60, C22×D15, C2×C4×D15
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, C22×C4, D10, C4×S3, C22×S3, D15, C4×D5, C22×D5, S3×C2×C4, D30, C2×C4×D5, C4×D15, C22×D15, C2×C4×D15
►On 120 points
Generators in S120
(1 73)(2 74)(3 75)(4 61)(5 62)(6 63)(7 64)(8 65)(9 66)(10 67)(11 68)(12 69)(13 70)(14 71)(15 72)(16 87)(17 88)(18 89)(19 90)(20 76)(21 77)(22 78)(23 79)(24 80)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 102)(32 103)(33 104)(34 105)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(43 99)(44 100)(45 101)(46 118)(47 119)(48 120)(49 106)(50 107)(51 108)(52 109)(53 110)(54 111)(55 112)(56 113)(57 114)(58 115)(59 116)(60 117)
(1 46 28 43)(2 47 29 44)(3 48 30 45)(4 49 16 31)(5 50 17 32)(6 51 18 33)(7 52 19 34)(8 53 20 35)(9 54 21 36)(10 55 22 37)(11 56 23 38)(12 57 24 39)(13 58 25 40)(14 59 26 41)(15 60 27 42)(61 106 87 102)(62 107 88 103)(63 108 89 104)(64 109 90 105)(65 110 76 91)(66 111 77 92)(67 112 78 93)(68 113 79 94)(69 114 80 95)(70 115 81 96)(71 116 82 97)(72 117 83 98)(73 118 84 99)(74 119 85 100)(75 120 86 101)
(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)
(1 27)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)(13 30)(14 29)(15 28)(31 57)(32 56)(33 55)(34 54)(35 53)(36 52)(37 51)(38 50)(39 49)(40 48)(41 47)(42 46)(43 60)(44 59)(45 58)(61 80)(62 79)(63 78)(64 77)(65 76)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(91 110)(92 109)(93 108)(94 107)(95 106)(96 120)(97 119)(98 118)(99 117)(100 116)(101 115)(102 114)(103 113)(104 112)(105 111)
G:=sub<Sym(120)| (1,73)(2,74)(3,75)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,87)(17,88)(18,89)(19,90)(20,76)(21,77)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,102)(32,103)(33,104)(34,105)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,118)(47,119)(48,120)(49,106)(50,107)(51,108)(52,109)(53,110)(54,111)(55,112)(56,113)(57,114)(58,115)(59,116)(60,117), (1,46,28,43)(2,47,29,44)(3,48,30,45)(4,49,16,31)(5,50,17,32)(6,51,18,33)(7,52,19,34)(8,53,20,35)(9,54,21,36)(10,55,22,37)(11,56,23,38)(12,57,24,39)(13,58,25,40)(14,59,26,41)(15,60,27,42)(61,106,87,102)(62,107,88,103)(63,108,89,104)(64,109,90,105)(65,110,76,91)(66,111,77,92)(67,112,78,93)(68,113,79,94)(69,114,80,95)(70,115,81,96)(71,116,82,97)(72,117,83,98)(73,118,84,99)(74,119,85,100)(75,120,86,101), (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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,30)(14,29)(15,28)(31,57)(32,56)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,48)(41,47)(42,46)(43,60)(44,59)(45,58)(61,80)(62,79)(63,78)(64,77)(65,76)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(91,110)(92,109)(93,108)(94,107)(95,106)(96,120)(97,119)(98,118)(99,117)(100,116)(101,115)(102,114)(103,113)(104,112)(105,111)>;
G:=Group( (1,73)(2,74)(3,75)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,87)(17,88)(18,89)(19,90)(20,76)(21,77)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,102)(32,103)(33,104)(34,105)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,118)(47,119)(48,120)(49,106)(50,107)(51,108)(52,109)(53,110)(54,111)(55,112)(56,113)(57,114)(58,115)(59,116)(60,117), (1,46,28,43)(2,47,29,44)(3,48,30,45)(4,49,16,31)(5,50,17,32)(6,51,18,33)(7,52,19,34)(8,53,20,35)(9,54,21,36)(10,55,22,37)(11,56,23,38)(12,57,24,39)(13,58,25,40)(14,59,26,41)(15,60,27,42)(61,106,87,102)(62,107,88,103)(63,108,89,104)(64,109,90,105)(65,110,76,91)(66,111,77,92)(67,112,78,93)(68,113,79,94)(69,114,80,95)(70,115,81,96)(71,116,82,97)(72,117,83,98)(73,118,84,99)(74,119,85,100)(75,120,86,101), (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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,30)(14,29)(15,28)(31,57)(32,56)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,48)(41,47)(42,46)(43,60)(44,59)(45,58)(61,80)(62,79)(63,78)(64,77)(65,76)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(91,110)(92,109)(93,108)(94,107)(95,106)(96,120)(97,119)(98,118)(99,117)(100,116)(101,115)(102,114)(103,113)(104,112)(105,111) );
G=PermutationGroup([[(1,73),(2,74),(3,75),(4,61),(5,62),(6,63),(7,64),(8,65),(9,66),(10,67),(11,68),(12,69),(13,70),(14,71),(15,72),(16,87),(17,88),(18,89),(19,90),(20,76),(21,77),(22,78),(23,79),(24,80),(25,81),(26,82),(27,83),(28,84),(29,85),(30,86),(31,102),(32,103),(33,104),(34,105),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,97),(42,98),(43,99),(44,100),(45,101),(46,118),(47,119),(48,120),(49,106),(50,107),(51,108),(52,109),(53,110),(54,111),(55,112),(56,113),(57,114),(58,115),(59,116),(60,117)], [(1,46,28,43),(2,47,29,44),(3,48,30,45),(4,49,16,31),(5,50,17,32),(6,51,18,33),(7,52,19,34),(8,53,20,35),(9,54,21,36),(10,55,22,37),(11,56,23,38),(12,57,24,39),(13,58,25,40),(14,59,26,41),(15,60,27,42),(61,106,87,102),(62,107,88,103),(63,108,89,104),(64,109,90,105),(65,110,76,91),(66,111,77,92),(67,112,78,93),(68,113,79,94),(69,114,80,95),(70,115,81,96),(71,116,82,97),(72,117,83,98),(73,118,84,99),(74,119,85,100),(75,120,86,101)], [(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)], [(1,27),(2,26),(3,25),(4,24),(5,23),(6,22),(7,21),(8,20),(9,19),(10,18),(11,17),(12,16),(13,30),(14,29),(15,28),(31,57),(32,56),(33,55),(34,54),(35,53),(36,52),(37,51),(38,50),(39,49),(40,48),(41,47),(42,46),(43,60),(44,59),(45,58),(61,80),(62,79),(63,78),(64,77),(65,76),(66,90),(67,89),(68,88),(69,87),(70,86),(71,85),(72,84),(73,83),(74,82),(75,81),(91,110),(92,109),(93,108),(94,107),(95,106),(96,120),(97,119),(98,118),(99,117),(100,116),(101,115),(102,114),(103,113),(104,112),(105,111)]])
C2×C4×D15 is a maximal subgroup of
D30⋊4C8 D30⋊3C8 D15⋊4M4(2) (C4×D15)⋊8C4 D30.34D4 D30.35D4 D30⋊8Q8 D30⋊9Q8 (C4×D15)⋊10C4 D30⋊10Q8 Dic15⋊13D4 D30.Q8 Dic15⋊14D4 D30.2Q8 D30⋊12D4 C12⋊2D20 C20⋊2D12 D30.27D4 C42⋊2D15 Dic15⋊19D4 D30.28D4 D30⋊9D4 C4⋊C4⋊7D15 D60⋊11C4 D30.29D4 C4⋊D60 D30⋊5Q8 D30⋊6Q8 C60⋊2D4 D30⋊7Q8 S3×C2×C4×D5 D20⋊24D6
C2×C4×D15 is a maximal quotient of
C42⋊2D15 C23.15D30 Dic15⋊19D4 Dic15⋊10Q8 C4⋊C4⋊7D15 D60⋊11C4 D60.6C4 D60.3C4
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 15 | 15 | 15 | 15 | 2 | 1 | 1 | 1 | 1 | 15 | 15 | 15 | 15 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D5 | D6 | D6 | D10 | D10 | C4×S3 | D15 | C4×D5 | D30 | D30 | C4×D15 |
| kernel | C2×C4×D15 | C4×D15 | C2×Dic15 | C2×C60 | C22×D15 | D30 | C2×C20 | C2×C12 | C20 | C2×C10 | C12 | C2×C6 | C10 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 1 | 4 | 2 | 4 | 4 | 8 | 8 | 4 | 16 |
Matrix representation of C2×C4×D15 ►in GL3(𝔽61) generated by
| 60 | 0 | 0 |
| 0 | 60 | 0 |
| 0 | 0 | 60 |
| 1 | 0 | 0 |
| 0 | 11 | 0 |
| 0 | 0 | 11 |
| 1 | 0 | 0 |
| 0 | 37 | 14 |
| 0 | 47 | 31 |
| 60 | 0 | 0 |
| 0 | 60 | 17 |
| 0 | 0 | 1 |
G:=sub<GL(3,GF(61))| [60,0,0,0,60,0,0,0,60],[1,0,0,0,11,0,0,0,11],[1,0,0,0,37,47,0,14,31],[60,0,0,0,60,0,0,17,1] >;
C2×C4×D15 in GAP, Magma, Sage, TeX
C_2\times C_4\times D_{15} % in TeX
G:=Group("C2xC4xD15"); // GroupNames label
G:=SmallGroup(240,176);
// by ID
G=gap.SmallGroup(240,176);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,50,964,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^15=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations