metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C60)⋊4C4, (C2×C12)⋊6F5, C60⋊C4⋊7C2, (D5×C12)⋊10C4, C60.60(C2×C4), (C4×D5)⋊5Dic3, (C4×D5).92D6, (C2×C20)⋊4Dic3, C12.53(C2×F5), (C6×Dic5)⋊15C4, C5⋊(C23.26D6), C15⋊4(C42⋊C2), C6.37(C22×F5), D5.5(C4○D12), (C2×Dic5)⋊9Dic3, C30.75(C22×C4), C20.21(C2×Dic3), (C6×D5).62C23, D10.17(C2×Dic3), D10.47(C22×S3), (C22×D5).103D6, C10.6(C22×Dic3), Dic5.17(C2×Dic3), D10.D6.2C2, (D5×C12).126C22, C3⋊4(D10.C23), (C4×C3⋊F5)⋊9C2, (C2×C4)⋊4(C3⋊F5), C4.21(C2×C3⋊F5), (C2×C4×D5).17S3, C22.7(C2×C3⋊F5), C2.7(C22×C3⋊F5), (D5×C2×C12).20C2, (C2×C6).48(C2×F5), (C2×C30).42(C2×C4), (C6×D5).60(C2×C4), (C2×C3⋊F5).15C22, (D5×C2×C6).145C22, (C3×D5).10(C4○D4), (C3×Dic5).67(C2×C4), (C2×C10).18(C2×Dic3), SmallGroup(480,1065)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 716 in 152 conjugacy classes, 61 normal (47 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4 [×6], C22, C22 [×4], C5, C6, C6 [×4], C2×C4, C2×C4 [×9], C23, D5 [×2], D5, C10, C10, Dic3 [×4], C12 [×2], C12 [×2], C2×C6, C2×C6 [×4], C15, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, Dic5 [×2], C20 [×2], F5 [×4], D10 [×2], D10 [×2], C2×C10, C2×Dic3 [×4], C2×C12, C2×C12 [×5], C22×C6, C3×D5 [×2], C3×D5, C30, C30, C42⋊C2, C4×D5 [×4], C2×Dic5, C2×C20, C2×F5 [×4], C22×D5, C4×Dic3 [×2], C4⋊Dic3 [×2], C6.D4 [×2], C22×C12, C3×Dic5 [×2], C60 [×2], C3⋊F5 [×4], C6×D5 [×2], C6×D5 [×2], C2×C30, C4×F5 [×2], C4⋊F5 [×2], C22⋊F5 [×2], C2×C4×D5, C23.26D6, D5×C12 [×4], C6×Dic5, C2×C60, C2×C3⋊F5 [×4], D5×C2×C6, D10.C23, C4×C3⋊F5 [×2], C60⋊C4 [×2], D10.D6 [×2], D5×C2×C12, (C2×C12)⋊6F5
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, Dic3 [×4], D6 [×3], C22×C4, C4○D4 [×2], F5, C2×Dic3 [×6], C22×S3, C42⋊C2, C2×F5 [×3], C4○D12 [×2], C22×Dic3, C3⋊F5, C22×F5, C23.26D6, C2×C3⋊F5 [×3], D10.C23, C22×C3⋊F5, (C2×C12)⋊6F5
Generators and relations
G = < a,b,c,d | a2=b12=c5=d4=1, ab=ba, ac=ca, dad-1=ab6, bc=cb, dbd-1=b5, dcd-1=c3 >
►On 120 points
(1 24)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(12 23)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)(33 37)(34 38)(35 39)(36 40)(49 93)(50 94)(51 95)(52 96)(53 85)(54 86)(55 87)(56 88)(57 89)(58 90)(59 91)(60 92)(61 114)(62 115)(63 116)(64 117)(65 118)(66 119)(67 120)(68 109)(69 110)(70 111)(71 112)(72 113)(73 104)(74 105)(75 106)(76 107)(77 108)(78 97)(79 98)(80 99)(81 100)(82 101)(83 102)(84 103)
(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 40 114 73 89)(2 41 115 74 90)(3 42 116 75 91)(4 43 117 76 92)(5 44 118 77 93)(6 45 119 78 94)(7 46 120 79 95)(8 47 109 80 96)(9 48 110 81 85)(10 37 111 82 86)(11 38 112 83 87)(12 39 113 84 88)(13 25 62 105 58)(14 26 63 106 59)(15 27 64 107 60)(16 28 65 108 49)(17 29 66 97 50)(18 30 67 98 51)(19 31 68 99 52)(20 32 69 100 53)(21 33 70 101 54)(22 34 71 102 55)(23 35 72 103 56)(24 36 61 104 57)
(2 6)(3 11)(5 9)(8 12)(13 23)(14 16)(15 21)(17 19)(18 24)(20 22)(25 72 58 103)(26 65 59 108)(27 70 60 101)(28 63 49 106)(29 68 50 99)(30 61 51 104)(31 66 52 97)(32 71 53 102)(33 64 54 107)(34 69 55 100)(35 62 56 105)(36 67 57 98)(37 111 86 82)(38 116 87 75)(39 109 88 80)(40 114 89 73)(41 119 90 78)(42 112 91 83)(43 117 92 76)(44 110 93 81)(45 115 94 74)(46 120 95 79)(47 113 96 84)(48 118 85 77)
G:=sub<Sym(120)| (1,24)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,37)(34,38)(35,39)(36,40)(49,93)(50,94)(51,95)(52,96)(53,85)(54,86)(55,87)(56,88)(57,89)(58,90)(59,91)(60,92)(61,114)(62,115)(63,116)(64,117)(65,118)(66,119)(67,120)(68,109)(69,110)(70,111)(71,112)(72,113)(73,104)(74,105)(75,106)(76,107)(77,108)(78,97)(79,98)(80,99)(81,100)(82,101)(83,102)(84,103), (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), (1,40,114,73,89)(2,41,115,74,90)(3,42,116,75,91)(4,43,117,76,92)(5,44,118,77,93)(6,45,119,78,94)(7,46,120,79,95)(8,47,109,80,96)(9,48,110,81,85)(10,37,111,82,86)(11,38,112,83,87)(12,39,113,84,88)(13,25,62,105,58)(14,26,63,106,59)(15,27,64,107,60)(16,28,65,108,49)(17,29,66,97,50)(18,30,67,98,51)(19,31,68,99,52)(20,32,69,100,53)(21,33,70,101,54)(22,34,71,102,55)(23,35,72,103,56)(24,36,61,104,57), (2,6)(3,11)(5,9)(8,12)(13,23)(14,16)(15,21)(17,19)(18,24)(20,22)(25,72,58,103)(26,65,59,108)(27,70,60,101)(28,63,49,106)(29,68,50,99)(30,61,51,104)(31,66,52,97)(32,71,53,102)(33,64,54,107)(34,69,55,100)(35,62,56,105)(36,67,57,98)(37,111,86,82)(38,116,87,75)(39,109,88,80)(40,114,89,73)(41,119,90,78)(42,112,91,83)(43,117,92,76)(44,110,93,81)(45,115,94,74)(46,120,95,79)(47,113,96,84)(48,118,85,77)>;
G:=Group( (1,24)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,37)(34,38)(35,39)(36,40)(49,93)(50,94)(51,95)(52,96)(53,85)(54,86)(55,87)(56,88)(57,89)(58,90)(59,91)(60,92)(61,114)(62,115)(63,116)(64,117)(65,118)(66,119)(67,120)(68,109)(69,110)(70,111)(71,112)(72,113)(73,104)(74,105)(75,106)(76,107)(77,108)(78,97)(79,98)(80,99)(81,100)(82,101)(83,102)(84,103), (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), (1,40,114,73,89)(2,41,115,74,90)(3,42,116,75,91)(4,43,117,76,92)(5,44,118,77,93)(6,45,119,78,94)(7,46,120,79,95)(8,47,109,80,96)(9,48,110,81,85)(10,37,111,82,86)(11,38,112,83,87)(12,39,113,84,88)(13,25,62,105,58)(14,26,63,106,59)(15,27,64,107,60)(16,28,65,108,49)(17,29,66,97,50)(18,30,67,98,51)(19,31,68,99,52)(20,32,69,100,53)(21,33,70,101,54)(22,34,71,102,55)(23,35,72,103,56)(24,36,61,104,57), (2,6)(3,11)(5,9)(8,12)(13,23)(14,16)(15,21)(17,19)(18,24)(20,22)(25,72,58,103)(26,65,59,108)(27,70,60,101)(28,63,49,106)(29,68,50,99)(30,61,51,104)(31,66,52,97)(32,71,53,102)(33,64,54,107)(34,69,55,100)(35,62,56,105)(36,67,57,98)(37,111,86,82)(38,116,87,75)(39,109,88,80)(40,114,89,73)(41,119,90,78)(42,112,91,83)(43,117,92,76)(44,110,93,81)(45,115,94,74)(46,120,95,79)(47,113,96,84)(48,118,85,77) );
G=PermutationGroup([(1,24),(2,13),(3,14),(4,15),(5,16),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22),(12,23),(25,41),(26,42),(27,43),(28,44),(29,45),(30,46),(31,47),(32,48),(33,37),(34,38),(35,39),(36,40),(49,93),(50,94),(51,95),(52,96),(53,85),(54,86),(55,87),(56,88),(57,89),(58,90),(59,91),(60,92),(61,114),(62,115),(63,116),(64,117),(65,118),(66,119),(67,120),(68,109),(69,110),(70,111),(71,112),(72,113),(73,104),(74,105),(75,106),(76,107),(77,108),(78,97),(79,98),(80,99),(81,100),(82,101),(83,102),(84,103)], [(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)], [(1,40,114,73,89),(2,41,115,74,90),(3,42,116,75,91),(4,43,117,76,92),(5,44,118,77,93),(6,45,119,78,94),(7,46,120,79,95),(8,47,109,80,96),(9,48,110,81,85),(10,37,111,82,86),(11,38,112,83,87),(12,39,113,84,88),(13,25,62,105,58),(14,26,63,106,59),(15,27,64,107,60),(16,28,65,108,49),(17,29,66,97,50),(18,30,67,98,51),(19,31,68,99,52),(20,32,69,100,53),(21,33,70,101,54),(22,34,71,102,55),(23,35,72,103,56),(24,36,61,104,57)], [(2,6),(3,11),(5,9),(8,12),(13,23),(14,16),(15,21),(17,19),(18,24),(20,22),(25,72,58,103),(26,65,59,108),(27,70,60,101),(28,63,49,106),(29,68,50,99),(30,61,51,104),(31,66,52,97),(32,71,53,102),(33,64,54,107),(34,69,55,100),(35,62,56,105),(36,67,57,98),(37,111,86,82),(38,116,87,75),(39,109,88,80),(40,114,89,73),(41,119,90,78),(42,112,91,83),(43,117,92,76),(44,110,93,81),(45,115,94,74),(46,120,95,79),(47,113,96,84),(48,118,85,77)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 7 | 0 | 14 | 14 |
| 47 | 54 | 47 | 0 |
| 0 | 47 | 54 | 47 |
| 14 | 14 | 0 | 7 |
| 8 | 0 | 5 | 5 |
| 56 | 3 | 56 | 0 |
| 0 | 56 | 3 | 56 |
| 5 | 5 | 0 | 8 |
| 60 | 60 | 60 | 60 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 60 | 0 | 0 |
| 1 | 1 | 1 | 1 |
G:=sub<GL(4,GF(61))| [7,47,0,14,0,54,47,14,14,47,54,0,14,0,47,7],[8,56,0,5,0,3,56,5,5,56,3,0,5,0,56,8],[60,1,0,0,60,0,1,0,60,0,0,1,60,0,0,0],[60,0,0,1,0,0,60,1,0,0,0,1,0,60,0,1] >;
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 5 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 5 | 5 | 10 | 2 | 1 | 1 | 2 | 5 | 5 | 10 | 30 | ··· | 30 | 4 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | - | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | Dic3 | D6 | Dic3 | Dic3 | D6 | C4○D4 | C4○D12 | F5 | C2×F5 | C2×F5 | C3⋊F5 | C2×C3⋊F5 | C2×C3⋊F5 | D10.C23 | (C2×C12)⋊6F5 |
| kernel | (C2×C12)⋊6F5 | C4×C3⋊F5 | C60⋊C4 | D10.D6 | D5×C2×C12 | D5×C12 | C6×Dic5 | C2×C60 | C2×C4×D5 | C4×D5 | C4×D5 | C2×Dic5 | C2×C20 | C22×D5 | C3×D5 | D5 | C2×C12 | C12 | C2×C6 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 8 | 1 | 2 | 1 | 2 | 4 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
(C_2\times C_{12})\rtimes_6F_5 % in TeX
G:=Group("(C2xC12):6F5"); // GroupNames label
G:=SmallGroup(480,1065);
// by ID
G=gap.SmallGroup(480,1065);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,422,2693,14118,2379]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^5=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^6,b*c=c*b,d*b*d^-1=b^5,d*c*d^-1=c^3>;
// generators/relations