metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.97D4, D20⋊5Dic3, C12.56D20, Dic10⋊5Dic3, C15⋊6C4≀C2, (C3×D20)⋊8C4, C12.8(C4×D5), C60.94(C2×C4), (C4×Dic3)⋊1D5, C4○D20.1S3, (C2×C30).24D4, C60.7C4⋊8C2, C4.3(D5×Dic3), C3⋊3(D20⋊4C4), (Dic3×C20)⋊1C2, (C3×Dic10)⋊8C4, (C2×C20).308D6, (C2×C12).56D10, C5⋊4(Q8⋊3Dic3), C4.28(C3⋊D20), C20.61(C3⋊D4), (C2×C60).36C22, C20.39(C2×Dic3), C30.60(C22⋊C4), C22.7(C15⋊D4), C6.31(D10⋊C4), C10.20(C6.D4), C2.10(D10⋊Dic3), (C2×C4).86(S3×D5), (C3×C4○D20).2C2, (C2×C6).1(C5⋊D4), (C2×C10).47(C3⋊D4), SmallGroup(480,53)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.97D4
G = < a,b,c | a60=b4=1, c2=a45, bab-1=a41, cac-1=a29, cbc-1=a45b-1 >
Subgroups: 364 in 88 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, C12, C2×C6, C2×C6, C15, C42, M4(2), C4○D4, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×Q8, C3×D5, C30, C30, C4≀C2, C5⋊2C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C4.Dic3, C4×Dic3, C3×C4○D4, C5×Dic3, C3×Dic5, C60, C6×D5, C2×C30, C4.Dic5, C4×C20, C4○D20, Q8⋊3Dic3, C15⋊3C8, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C10×Dic3, C2×C60, D20⋊4C4, Dic3×C20, C60.7C4, C3×C4○D20, C60.97D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, Dic3, D6, C22⋊C4, D10, C2×Dic3, C3⋊D4, C4≀C2, C4×D5, D20, C5⋊D4, C6.D4, S3×D5, D10⋊C4, Q8⋊3Dic3, D5×Dic3, C15⋊D4, C3⋊D20, D20⋊4C4, D10⋊Dic3, C60.97D4
►On 120 points
(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 31)(2 12)(3 53)(4 34)(5 15)(6 56)(7 37)(8 18)(9 59)(10 40)(11 21)(13 43)(14 24)(16 46)(17 27)(19 49)(20 30)(22 52)(23 33)(25 55)(26 36)(28 58)(29 39)(32 42)(35 45)(38 48)(41 51)(44 54)(47 57)(50 60)(61 76 91 106)(62 117 92 87)(63 98 93 68)(64 79 94 109)(65 120 95 90)(66 101 96 71)(67 82 97 112)(69 104 99 74)(70 85 100 115)(72 107 102 77)(73 88 103 118)(75 110 105 80)(78 113 108 83)(81 116 111 86)(84 119 114 89)
(1 76 46 61 31 106 16 91)(2 105 47 90 32 75 17 120)(3 74 48 119 33 104 18 89)(4 103 49 88 34 73 19 118)(5 72 50 117 35 102 20 87)(6 101 51 86 36 71 21 116)(7 70 52 115 37 100 22 85)(8 99 53 84 38 69 23 114)(9 68 54 113 39 98 24 83)(10 97 55 82 40 67 25 112)(11 66 56 111 41 96 26 81)(12 95 57 80 42 65 27 110)(13 64 58 109 43 94 28 79)(14 93 59 78 44 63 29 108)(15 62 60 107 45 92 30 77)
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), (1,31)(2,12)(3,53)(4,34)(5,15)(6,56)(7,37)(8,18)(9,59)(10,40)(11,21)(13,43)(14,24)(16,46)(17,27)(19,49)(20,30)(22,52)(23,33)(25,55)(26,36)(28,58)(29,39)(32,42)(35,45)(38,48)(41,51)(44,54)(47,57)(50,60)(61,76,91,106)(62,117,92,87)(63,98,93,68)(64,79,94,109)(65,120,95,90)(66,101,96,71)(67,82,97,112)(69,104,99,74)(70,85,100,115)(72,107,102,77)(73,88,103,118)(75,110,105,80)(78,113,108,83)(81,116,111,86)(84,119,114,89), (1,76,46,61,31,106,16,91)(2,105,47,90,32,75,17,120)(3,74,48,119,33,104,18,89)(4,103,49,88,34,73,19,118)(5,72,50,117,35,102,20,87)(6,101,51,86,36,71,21,116)(7,70,52,115,37,100,22,85)(8,99,53,84,38,69,23,114)(9,68,54,113,39,98,24,83)(10,97,55,82,40,67,25,112)(11,66,56,111,41,96,26,81)(12,95,57,80,42,65,27,110)(13,64,58,109,43,94,28,79)(14,93,59,78,44,63,29,108)(15,62,60,107,45,92,30,77)>;
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), (1,31)(2,12)(3,53)(4,34)(5,15)(6,56)(7,37)(8,18)(9,59)(10,40)(11,21)(13,43)(14,24)(16,46)(17,27)(19,49)(20,30)(22,52)(23,33)(25,55)(26,36)(28,58)(29,39)(32,42)(35,45)(38,48)(41,51)(44,54)(47,57)(50,60)(61,76,91,106)(62,117,92,87)(63,98,93,68)(64,79,94,109)(65,120,95,90)(66,101,96,71)(67,82,97,112)(69,104,99,74)(70,85,100,115)(72,107,102,77)(73,88,103,118)(75,110,105,80)(78,113,108,83)(81,116,111,86)(84,119,114,89), (1,76,46,61,31,106,16,91)(2,105,47,90,32,75,17,120)(3,74,48,119,33,104,18,89)(4,103,49,88,34,73,19,118)(5,72,50,117,35,102,20,87)(6,101,51,86,36,71,21,116)(7,70,52,115,37,100,22,85)(8,99,53,84,38,69,23,114)(9,68,54,113,39,98,24,83)(10,97,55,82,40,67,25,112)(11,66,56,111,41,96,26,81)(12,95,57,80,42,65,27,110)(13,64,58,109,43,94,28,79)(14,93,59,78,44,63,29,108)(15,62,60,107,45,92,30,77) );
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)], [(1,31),(2,12),(3,53),(4,34),(5,15),(6,56),(7,37),(8,18),(9,59),(10,40),(11,21),(13,43),(14,24),(16,46),(17,27),(19,49),(20,30),(22,52),(23,33),(25,55),(26,36),(28,58),(29,39),(32,42),(35,45),(38,48),(41,51),(44,54),(47,57),(50,60),(61,76,91,106),(62,117,92,87),(63,98,93,68),(64,79,94,109),(65,120,95,90),(66,101,96,71),(67,82,97,112),(69,104,99,74),(70,85,100,115),(72,107,102,77),(73,88,103,118),(75,110,105,80),(78,113,108,83),(81,116,111,86),(84,119,114,89)], [(1,76,46,61,31,106,16,91),(2,105,47,90,32,75,17,120),(3,74,48,119,33,104,18,89),(4,103,49,88,34,73,19,118),(5,72,50,117,35,102,20,87),(6,101,51,86,36,71,21,116),(7,70,52,115,37,100,22,85),(8,99,53,84,38,69,23,114),(9,68,54,113,39,98,24,83),(10,97,55,82,40,67,25,112),(11,66,56,111,41,96,26,81),(12,95,57,80,42,65,27,110),(13,64,58,109,43,94,28,79),(14,93,59,78,44,63,29,108),(15,62,60,107,45,92,30,77)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 6D | 8A | 8B | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20X | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 20 | 2 | 1 | 1 | 2 | 6 | 6 | 6 | 6 | 20 | 2 | 2 | 2 | 4 | 20 | 20 | 60 | 60 | 2 | ··· | 2 | 2 | 2 | 4 | 20 | 20 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | + | + | + | + | - | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | D5 | Dic3 | Dic3 | D6 | D10 | C3⋊D4 | C3⋊D4 | C4≀C2 | C4×D5 | D20 | C5⋊D4 | D20⋊4C4 | S3×D5 | Q8⋊3Dic3 | D5×Dic3 | C3⋊D20 | C15⋊D4 | C60.97D4 |
| kernel | C60.97D4 | Dic3×C20 | C60.7C4 | C3×C4○D20 | C3×Dic10 | C3×D20 | C4○D20 | C60 | C2×C30 | C4×Dic3 | Dic10 | D20 | C2×C20 | C2×C12 | C20 | C2×C10 | C15 | C12 | C12 | C2×C6 | C3 | C2×C4 | C5 | C4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 16 | 2 | 2 | 2 | 2 | 2 | 8 |
Matrix representation of C60.97D4 ►in GL4(𝔽241) generated by
| 240 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 |
| 0 | 0 | 135 | 0 |
| 0 | 0 | 0 | 216 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 64 |
| 0 | 240 | 0 | 0 |
| 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 64 |
| 0 | 0 | 240 | 0 |
G:=sub<GL(4,GF(241))| [240,240,0,0,1,0,0,0,0,0,135,0,0,0,0,216],[0,1,0,0,1,0,0,0,0,0,240,0,0,0,0,64],[0,240,0,0,240,0,0,0,0,0,0,240,0,0,64,0] >;
C60.97D4 in GAP, Magma, Sage, TeX
C_{60}._{97}D_4 % in TeX
G:=Group("C60.97D4"); // GroupNames label
G:=SmallGroup(480,53);
// by ID
G=gap.SmallGroup(480,53);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,36,422,675,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=1,c^2=a^45,b*a*b^-1=a^41,c*a*c^-1=a^29,c*b*c^-1=a^45*b^-1>;
// generators/relations