metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.38D4, D12⋊19D10, C20.51D12, Dic6⋊16D10, C60.129C23, D60.49C22, C5⋊2C8⋊4D6, C4○D12⋊4D5, C5⋊5(C8⋊D6), (C2×D60)⋊18C2, C5⋊D24⋊14C2, (C2×C20).91D6, (C2×C10).6D12, C30.80(C2×D4), (C2×C30).48D4, C4.Dic5⋊8S3, C3⋊1(D4⋊D10), C15⋊11(C8⋊C22), C10.50(C2×D12), (C2×C12).91D10, Dic6⋊D5⋊13C2, (C5×D12)⋊21C22, C12.28(C5⋊D4), C4.16(C5⋊D12), C20.91(C22×S3), (C2×C60).94C22, (C5×Dic6)⋊18C22, C12.152(C22×D5), C22.9(C5⋊D12), C4.77(C2×S3×D5), C6.4(C2×C5⋊D4), (C5×C4○D12)⋊6C2, (C2×C4).11(S3×D5), C2.8(C2×C5⋊D12), (C3×C4.Dic5)⋊8C2, (C3×C5⋊2C8)⋊18C22, (C2×C6).12(C5⋊D4), SmallGroup(480,381)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.38D4
G = < a,b,c | a60=c2=1, b4=a30, bab-1=a19, cac=a-1, cbc=a30b3 >
Subgroups: 956 in 136 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, C20, C20, D10, C2×C10, C2×C10, C24, Dic6, C4×S3, D12, D12, C3⋊D4, C2×C12, C22×S3, C5×S3, D15, C30, C30, C8⋊C22, C5⋊2C8, D20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C24⋊C2, D24, C3×M4(2), C2×D12, C4○D12, C5×Dic3, C60, S3×C10, D30, C2×C30, C4.Dic5, D4⋊D5, Q8⋊D5, C2×D20, C5×C4○D4, C8⋊D6, C3×C5⋊2C8, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, D60, D60, C2×C60, C22×D15, D4⋊D10, C5⋊D24, Dic6⋊D5, C3×C4.Dic5, C5×C4○D12, C2×D60, C60.38D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, D12, C22×S3, C8⋊C22, C5⋊D4, C22×D5, C2×D12, S3×D5, C2×C5⋊D4, C8⋊D6, C5⋊D12, C2×S3×D5, D4⋊D10, C2×C5⋊D12, C60.38D4
►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 75 16 120 31 105 46 90)(2 94 17 79 32 64 47 109)(3 113 18 98 33 83 48 68)(4 72 19 117 34 102 49 87)(5 91 20 76 35 61 50 106)(6 110 21 95 36 80 51 65)(7 69 22 114 37 99 52 84)(8 88 23 73 38 118 53 103)(9 107 24 92 39 77 54 62)(10 66 25 111 40 96 55 81)(11 85 26 70 41 115 56 100)(12 104 27 89 42 74 57 119)(13 63 28 108 43 93 58 78)(14 82 29 67 44 112 59 97)(15 101 30 86 45 71 60 116)
(2 60)(3 59)(4 58)(5 57)(6 56)(7 55)(8 54)(9 53)(10 52)(11 51)(12 50)(13 49)(14 48)(15 47)(16 46)(17 45)(18 44)(19 43)(20 42)(21 41)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(61 104)(62 103)(63 102)(64 101)(65 100)(66 99)(67 98)(68 97)(69 96)(70 95)(71 94)(72 93)(73 92)(74 91)(75 90)(76 89)(77 88)(78 87)(79 86)(80 85)(81 84)(82 83)(105 120)(106 119)(107 118)(108 117)(109 116)(110 115)(111 114)(112 113)
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), (1,75,16,120,31,105,46,90)(2,94,17,79,32,64,47,109)(3,113,18,98,33,83,48,68)(4,72,19,117,34,102,49,87)(5,91,20,76,35,61,50,106)(6,110,21,95,36,80,51,65)(7,69,22,114,37,99,52,84)(8,88,23,73,38,118,53,103)(9,107,24,92,39,77,54,62)(10,66,25,111,40,96,55,81)(11,85,26,70,41,115,56,100)(12,104,27,89,42,74,57,119)(13,63,28,108,43,93,58,78)(14,82,29,67,44,112,59,97)(15,101,30,86,45,71,60,116), (2,60)(3,59)(4,58)(5,57)(6,56)(7,55)(8,54)(9,53)(10,52)(11,51)(12,50)(13,49)(14,48)(15,47)(16,46)(17,45)(18,44)(19,43)(20,42)(21,41)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,98)(68,97)(69,96)(70,95)(71,94)(72,93)(73,92)(74,91)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(81,84)(82,83)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113)>;
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), (1,75,16,120,31,105,46,90)(2,94,17,79,32,64,47,109)(3,113,18,98,33,83,48,68)(4,72,19,117,34,102,49,87)(5,91,20,76,35,61,50,106)(6,110,21,95,36,80,51,65)(7,69,22,114,37,99,52,84)(8,88,23,73,38,118,53,103)(9,107,24,92,39,77,54,62)(10,66,25,111,40,96,55,81)(11,85,26,70,41,115,56,100)(12,104,27,89,42,74,57,119)(13,63,28,108,43,93,58,78)(14,82,29,67,44,112,59,97)(15,101,30,86,45,71,60,116), (2,60)(3,59)(4,58)(5,57)(6,56)(7,55)(8,54)(9,53)(10,52)(11,51)(12,50)(13,49)(14,48)(15,47)(16,46)(17,45)(18,44)(19,43)(20,42)(21,41)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,98)(68,97)(69,96)(70,95)(71,94)(72,93)(73,92)(74,91)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(81,84)(82,83)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113) );
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)], [(1,75,16,120,31,105,46,90),(2,94,17,79,32,64,47,109),(3,113,18,98,33,83,48,68),(4,72,19,117,34,102,49,87),(5,91,20,76,35,61,50,106),(6,110,21,95,36,80,51,65),(7,69,22,114,37,99,52,84),(8,88,23,73,38,118,53,103),(9,107,24,92,39,77,54,62),(10,66,25,111,40,96,55,81),(11,85,26,70,41,115,56,100),(12,104,27,89,42,74,57,119),(13,63,28,108,43,93,58,78),(14,82,29,67,44,112,59,97),(15,101,30,86,45,71,60,116)], [(2,60),(3,59),(4,58),(5,57),(6,56),(7,55),(8,54),(9,53),(10,52),(11,51),(12,50),(13,49),(14,48),(15,47),(16,46),(17,45),(18,44),(19,43),(20,42),(21,41),(22,40),(23,39),(24,38),(25,37),(26,36),(27,35),(28,34),(29,33),(30,32),(61,104),(62,103),(63,102),(64,101),(65,100),(66,99),(67,98),(68,97),(69,96),(70,95),(71,94),(72,93),(73,92),(74,91),(75,90),(76,89),(77,88),(78,87),(79,86),(80,85),(81,84),(82,83),(105,120),(106,119),(107,118),(108,117),(109,116),(110,115),(111,114),(112,113)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 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 | 2 | 2 | 3 | 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 | 60 | 60 | 2 | 2 | 2 | 12 | 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 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D10 | D10 | D10 | D12 | D12 | C5⋊D4 | C5⋊D4 | C8⋊C22 | S3×D5 | C8⋊D6 | C5⋊D12 | C2×S3×D5 | C5⋊D12 | D4⋊D10 | C60.38D4 |
| kernel | C60.38D4 | C5⋊D24 | Dic6⋊D5 | C3×C4.Dic5 | C5×C4○D12 | C2×D60 | C4.Dic5 | C60 | C2×C30 | C4○D12 | C5⋊2C8 | C2×C20 | Dic6 | D12 | C2×C12 | C20 | C2×C10 | C12 | C2×C6 | C15 | C2×C4 | C5 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of C60.38D4 ►in GL4(𝔽241) generated by
| 184 | 21 | 0 | 0 |
| 220 | 50 | 0 | 0 |
| 22 | 22 | 220 | 57 |
| 219 | 180 | 184 | 235 |
| 214 | 157 | 153 | 235 |
| 226 | 27 | 85 | 88 |
| 16 | 212 | 27 | 84 |
| 64 | 225 | 15 | 214 |
| 1 | 0 | 0 | 0 |
| 51 | 240 | 0 | 0 |
| 120 | 42 | 44 | 3 |
| 137 | 121 | 78 | 197 |
G:=sub<GL(4,GF(241))| [184,220,22,219,21,50,22,180,0,0,220,184,0,0,57,235],[214,226,16,64,157,27,212,225,153,85,27,15,235,88,84,214],[1,51,120,137,0,240,42,121,0,0,44,78,0,0,3,197] >;
C60.38D4 in GAP, Magma, Sage, TeX
C_{60}._{38}D_4 % in TeX
G:=Group("C60.38D4"); // GroupNames label
G:=SmallGroup(480,381);
// by ID
G=gap.SmallGroup(480,381);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,422,100,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=c^2=1,b^4=a^30,b*a*b^-1=a^19,c*a*c=a^-1,c*b*c=a^30*b^3>;
// generators/relations