metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.98D4, D12⋊4Dic5, Dic6⋊4Dic5, C15⋊7C4≀C2, (C5×D12)⋊11C4, C20.39(C4×S3), C4○D12.2D5, (C2×C10).1D12, (C2×C30).25D4, (C2×C20).55D6, C4.Dic5⋊4S3, C4.8(S3×Dic5), C5⋊5(D12⋊C4), C60.125(C2×C4), (C5×Dic6)⋊11C4, (C2×C12).57D10, C12.3(C2×Dic5), C10.45(D6⋊C4), (C4×Dic15)⋊29C2, C3⋊1(D4⋊2Dic5), C4.29(C15⋊D4), C12.87(C5⋊D4), C20.87(C3⋊D4), C2.9(D6⋊Dic5), C6.8(C23.D5), C30.61(C22⋊C4), (C2×C60).211C22, C22.7(C5⋊D12), (C5×C4○D12).5C2, (C2×C4).189(S3×D5), (C2×C6).2(C5⋊D4), (C3×C4.Dic5)⋊10C2, SmallGroup(480,54)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.98D4
G = < a,b,c,d | a12=c10=1, b2=a6, d2=c5, bab-1=a-1, ac=ca, dad-1=a5, cbc-1=a6b, dbd-1=a3b, dcd-1=c-1 >
Subgroups: 380 in 88 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C10, C10, Dic3, C12, D6, C2×C6, C15, C42, M4(2), C4○D4, Dic5, C20, C20, C2×C10, C2×C10, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C5×S3, C30, C30, C4≀C2, C5⋊2C8, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×Q8, C4×Dic3, C3×M4(2), C4○D12, C5×Dic3, Dic15, C60, S3×C10, C2×C30, C4.Dic5, C4×Dic5, C5×C4○D4, D12⋊C4, C3×C5⋊2C8, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, C2×Dic15, C2×C60, D4⋊2Dic5, C3×C4.Dic5, C4×Dic15, C5×C4○D12, C60.98D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, D6, C22⋊C4, Dic5, D10, C4×S3, D12, C3⋊D4, C4≀C2, C2×Dic5, C5⋊D4, D6⋊C4, S3×D5, C23.D5, D12⋊C4, S3×Dic5, C15⋊D4, C5⋊D12, D4⋊2Dic5, D6⋊Dic5, C60.98D4
►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 94 7 88)(2 93 8 87)(3 92 9 86)(4 91 10 85)(5 90 11 96)(6 89 12 95)(13 76 19 82)(14 75 20 81)(15 74 21 80)(16 73 22 79)(17 84 23 78)(18 83 24 77)(25 49 31 55)(26 60 32 54)(27 59 33 53)(28 58 34 52)(29 57 35 51)(30 56 36 50)(37 107 43 101)(38 106 44 100)(39 105 45 99)(40 104 46 98)(41 103 47 97)(42 102 48 108)(61 110 67 116)(62 109 68 115)(63 120 69 114)(64 119 70 113)(65 118 71 112)(66 117 72 111)
(1 47 84 70 51)(2 48 73 71 52)(3 37 74 72 53)(4 38 75 61 54)(5 39 76 62 55)(6 40 77 63 56)(7 41 78 64 57)(8 42 79 65 58)(9 43 80 66 59)(10 44 81 67 60)(11 45 82 68 49)(12 46 83 69 50)(13 109 31 90 99 19 115 25 96 105)(14 110 32 91 100 20 116 26 85 106)(15 111 33 92 101 21 117 27 86 107)(16 112 34 93 102 22 118 28 87 108)(17 113 35 94 103 23 119 29 88 97)(18 114 36 95 104 24 120 30 89 98)
(1 7)(2 12)(3 5)(4 10)(6 8)(9 11)(13 120 19 114)(14 113 20 119)(15 118 21 112)(16 111 22 117)(17 116 23 110)(18 109 24 115)(25 98 31 104)(26 103 32 97)(27 108 33 102)(28 101 34 107)(29 106 35 100)(30 99 36 105)(37 55)(38 60)(39 53)(40 58)(41 51)(42 56)(43 49)(44 54)(45 59)(46 52)(47 57)(48 50)(61 81)(62 74)(63 79)(64 84)(65 77)(66 82)(67 75)(68 80)(69 73)(70 78)(71 83)(72 76)(85 94 91 88)(86 87 92 93)(89 90 95 96)
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), (1,94,7,88)(2,93,8,87)(3,92,9,86)(4,91,10,85)(5,90,11,96)(6,89,12,95)(13,76,19,82)(14,75,20,81)(15,74,21,80)(16,73,22,79)(17,84,23,78)(18,83,24,77)(25,49,31,55)(26,60,32,54)(27,59,33,53)(28,58,34,52)(29,57,35,51)(30,56,36,50)(37,107,43,101)(38,106,44,100)(39,105,45,99)(40,104,46,98)(41,103,47,97)(42,102,48,108)(61,110,67,116)(62,109,68,115)(63,120,69,114)(64,119,70,113)(65,118,71,112)(66,117,72,111), (1,47,84,70,51)(2,48,73,71,52)(3,37,74,72,53)(4,38,75,61,54)(5,39,76,62,55)(6,40,77,63,56)(7,41,78,64,57)(8,42,79,65,58)(9,43,80,66,59)(10,44,81,67,60)(11,45,82,68,49)(12,46,83,69,50)(13,109,31,90,99,19,115,25,96,105)(14,110,32,91,100,20,116,26,85,106)(15,111,33,92,101,21,117,27,86,107)(16,112,34,93,102,22,118,28,87,108)(17,113,35,94,103,23,119,29,88,97)(18,114,36,95,104,24,120,30,89,98), (1,7)(2,12)(3,5)(4,10)(6,8)(9,11)(13,120,19,114)(14,113,20,119)(15,118,21,112)(16,111,22,117)(17,116,23,110)(18,109,24,115)(25,98,31,104)(26,103,32,97)(27,108,33,102)(28,101,34,107)(29,106,35,100)(30,99,36,105)(37,55)(38,60)(39,53)(40,58)(41,51)(42,56)(43,49)(44,54)(45,59)(46,52)(47,57)(48,50)(61,81)(62,74)(63,79)(64,84)(65,77)(66,82)(67,75)(68,80)(69,73)(70,78)(71,83)(72,76)(85,94,91,88)(86,87,92,93)(89,90,95,96)>;
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), (1,94,7,88)(2,93,8,87)(3,92,9,86)(4,91,10,85)(5,90,11,96)(6,89,12,95)(13,76,19,82)(14,75,20,81)(15,74,21,80)(16,73,22,79)(17,84,23,78)(18,83,24,77)(25,49,31,55)(26,60,32,54)(27,59,33,53)(28,58,34,52)(29,57,35,51)(30,56,36,50)(37,107,43,101)(38,106,44,100)(39,105,45,99)(40,104,46,98)(41,103,47,97)(42,102,48,108)(61,110,67,116)(62,109,68,115)(63,120,69,114)(64,119,70,113)(65,118,71,112)(66,117,72,111), (1,47,84,70,51)(2,48,73,71,52)(3,37,74,72,53)(4,38,75,61,54)(5,39,76,62,55)(6,40,77,63,56)(7,41,78,64,57)(8,42,79,65,58)(9,43,80,66,59)(10,44,81,67,60)(11,45,82,68,49)(12,46,83,69,50)(13,109,31,90,99,19,115,25,96,105)(14,110,32,91,100,20,116,26,85,106)(15,111,33,92,101,21,117,27,86,107)(16,112,34,93,102,22,118,28,87,108)(17,113,35,94,103,23,119,29,88,97)(18,114,36,95,104,24,120,30,89,98), (1,7)(2,12)(3,5)(4,10)(6,8)(9,11)(13,120,19,114)(14,113,20,119)(15,118,21,112)(16,111,22,117)(17,116,23,110)(18,109,24,115)(25,98,31,104)(26,103,32,97)(27,108,33,102)(28,101,34,107)(29,106,35,100)(30,99,36,105)(37,55)(38,60)(39,53)(40,58)(41,51)(42,56)(43,49)(44,54)(45,59)(46,52)(47,57)(48,50)(61,81)(62,74)(63,79)(64,84)(65,77)(66,82)(67,75)(68,80)(69,73)(70,78)(71,83)(72,76)(85,94,91,88)(86,87,92,93)(89,90,95,96) );
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)], [(1,94,7,88),(2,93,8,87),(3,92,9,86),(4,91,10,85),(5,90,11,96),(6,89,12,95),(13,76,19,82),(14,75,20,81),(15,74,21,80),(16,73,22,79),(17,84,23,78),(18,83,24,77),(25,49,31,55),(26,60,32,54),(27,59,33,53),(28,58,34,52),(29,57,35,51),(30,56,36,50),(37,107,43,101),(38,106,44,100),(39,105,45,99),(40,104,46,98),(41,103,47,97),(42,102,48,108),(61,110,67,116),(62,109,68,115),(63,120,69,114),(64,119,70,113),(65,118,71,112),(66,117,72,111)], [(1,47,84,70,51),(2,48,73,71,52),(3,37,74,72,53),(4,38,75,61,54),(5,39,76,62,55),(6,40,77,63,56),(7,41,78,64,57),(8,42,79,65,58),(9,43,80,66,59),(10,44,81,67,60),(11,45,82,68,49),(12,46,83,69,50),(13,109,31,90,99,19,115,25,96,105),(14,110,32,91,100,20,116,26,85,106),(15,111,33,92,101,21,117,27,86,107),(16,112,34,93,102,22,118,28,87,108),(17,113,35,94,103,23,119,29,88,97),(18,114,36,95,104,24,120,30,89,98)], [(1,7),(2,12),(3,5),(4,10),(6,8),(9,11),(13,120,19,114),(14,113,20,119),(15,118,21,112),(16,111,22,117),(17,116,23,110),(18,109,24,115),(25,98,31,104),(26,103,32,97),(27,108,33,102),(28,101,34,107),(29,106,35,100),(30,99,36,105),(37,55),(38,60),(39,53),(40,58),(41,51),(42,56),(43,49),(44,54),(45,59),(46,52),(47,57),(48,50),(61,81),(62,74),(63,79),(64,84),(65,77),(66,82),(67,75),(68,80),(69,73),(70,78),(71,83),(72,76),(85,94,91,88),(86,87,92,93),(89,90,95,96)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 12C | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 24A | 24B | 24C | 24D | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 12 | 2 | 1 | 1 | 2 | 12 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 4 | 20 | 20 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 20 | 20 | 20 | 20 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | + | + | + | - | - | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | D5 | D6 | Dic5 | Dic5 | D10 | C4×S3 | C3⋊D4 | D12 | C4≀C2 | C5⋊D4 | C5⋊D4 | S3×D5 | D12⋊C4 | S3×Dic5 | C15⋊D4 | C5⋊D12 | D4⋊2Dic5 | C60.98D4 |
| kernel | C60.98D4 | C3×C4.Dic5 | C4×Dic15 | C5×C4○D12 | C5×Dic6 | C5×D12 | C4.Dic5 | C60 | C2×C30 | C4○D12 | C2×C20 | Dic6 | D12 | C2×C12 | C20 | C20 | C2×C10 | C15 | C12 | C2×C6 | C2×C4 | C5 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of C60.98D4 ►in GL6(𝔽241)
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 | 0 |
| 0 | 0 | 0 | 177 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 1 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 76 | 89 | 0 | 0 | 0 | 0 |
| 103 | 165 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 177 | 0 | 0 |
| 0 | 0 | 177 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 240 | 191 | 0 | 0 | 0 | 0 |
| 240 | 190 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 51 | 191 | 0 | 0 | 0 | 0 |
| 52 | 190 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 177 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,64,0,0,0,0,0,0,177,0,0,0,0,0,0,240,240,0,0,0,0,1,0],[76,103,0,0,0,0,89,165,0,0,0,0,0,0,0,177,0,0,0,0,177,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[240,240,0,0,0,0,191,190,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[51,52,0,0,0,0,191,190,0,0,0,0,0,0,240,0,0,0,0,0,0,177,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C60.98D4 in GAP, Magma, Sage, TeX
C_{60}._{98}D_4 % in TeX
G:=Group("C60.98D4"); // GroupNames label
G:=SmallGroup(480,54);
// by ID
G=gap.SmallGroup(480,54);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,100,675,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^10=1,b^2=a^6,d^2=c^5,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^5,c*b*c^-1=a^6*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations