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), (C2×Dic5)⋊9Dic3, D5.5(C4○D12), 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), D10.D6.2C2, Dic5.17(C2×Dic3), (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, C2.7(C22×C3⋊F5), C22.7(C2×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
Generators and relations for (C2×C12)⋊6F5
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 >
Subgroups: 716 in 152 conjugacy classes, 61 normal (47 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, D5, D5, C10, C10, Dic3, C12, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C2×Dic3, C2×C12, C2×C12, C22×C6, C3×D5, C3×D5, C30, C30, C42⋊C2, C4×D5, C2×Dic5, C2×C20, C2×F5, C22×D5, C4×Dic3, C4⋊Dic3, C6.D4, C22×C12, C3×Dic5, C60, C3⋊F5, C6×D5, C6×D5, C2×C30, C4×F5, C4⋊F5, C22⋊F5, C2×C4×D5, C23.26D6, D5×C12, C6×Dic5, C2×C60, C2×C3⋊F5, D5×C2×C6, D10.C23, C4×C3⋊F5, C60⋊C4, D10.D6, D5×C2×C12, (C2×C12)⋊6F5
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C4○D4, F5, C2×Dic3, C22×S3, C42⋊C2, C2×F5, C4○D12, C22×Dic3, C3⋊F5, C22×F5, C23.26D6, C2×C3⋊F5, D10.C23, C22×C3⋊F5, (C2×C12)⋊6F5
►On 120 points
Generators in S120
(1 110)(2 111)(3 112)(4 113)(5 114)(6 115)(7 116)(8 117)(9 118)(10 119)(11 120)(12 109)(13 106)(14 107)(15 108)(16 97)(17 98)(18 99)(19 100)(20 101)(21 102)(22 103)(23 104)(24 105)(25 66)(26 67)(27 68)(28 69)(29 70)(30 71)(31 72)(32 61)(33 62)(34 63)(35 64)(36 65)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 87)(50 88)(51 89)(52 90)(53 91)(54 92)(55 93)(56 94)(57 95)(58 96)(59 85)(60 86)
(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 71 90 84 100)(2 72 91 73 101)(3 61 92 74 102)(4 62 93 75 103)(5 63 94 76 104)(6 64 95 77 105)(7 65 96 78 106)(8 66 85 79 107)(9 67 86 80 108)(10 68 87 81 97)(11 69 88 82 98)(12 70 89 83 99)(13 116 36 58 48)(14 117 25 59 37)(15 118 26 60 38)(16 119 27 49 39)(17 120 28 50 40)(18 109 29 51 41)(19 110 30 52 42)(20 111 31 53 43)(21 112 32 54 44)(22 113 33 55 45)(23 114 34 56 46)(24 115 35 57 47)
(2 6)(3 11)(5 9)(8 12)(13 42 36 52)(14 47 25 57)(15 40 26 50)(16 45 27 55)(17 38 28 60)(18 43 29 53)(19 48 30 58)(20 41 31 51)(21 46 32 56)(22 39 33 49)(23 44 34 54)(24 37 35 59)(61 88 102 82)(62 93 103 75)(63 86 104 80)(64 91 105 73)(65 96 106 78)(66 89 107 83)(67 94 108 76)(68 87 97 81)(69 92 98 74)(70 85 99 79)(71 90 100 84)(72 95 101 77)(109 111)(110 116)(112 114)(113 119)(115 117)(118 120)
G:=sub<Sym(120)| (1,110)(2,111)(3,112)(4,113)(5,114)(6,115)(7,116)(8,117)(9,118)(10,119)(11,120)(12,109)(13,106)(14,107)(15,108)(16,97)(17,98)(18,99)(19,100)(20,101)(21,102)(22,103)(23,104)(24,105)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,61)(33,62)(34,63)(35,64)(36,65)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,87)(50,88)(51,89)(52,90)(53,91)(54,92)(55,93)(56,94)(57,95)(58,96)(59,85)(60,86), (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), (1,71,90,84,100)(2,72,91,73,101)(3,61,92,74,102)(4,62,93,75,103)(5,63,94,76,104)(6,64,95,77,105)(7,65,96,78,106)(8,66,85,79,107)(9,67,86,80,108)(10,68,87,81,97)(11,69,88,82,98)(12,70,89,83,99)(13,116,36,58,48)(14,117,25,59,37)(15,118,26,60,38)(16,119,27,49,39)(17,120,28,50,40)(18,109,29,51,41)(19,110,30,52,42)(20,111,31,53,43)(21,112,32,54,44)(22,113,33,55,45)(23,114,34,56,46)(24,115,35,57,47), (2,6)(3,11)(5,9)(8,12)(13,42,36,52)(14,47,25,57)(15,40,26,50)(16,45,27,55)(17,38,28,60)(18,43,29,53)(19,48,30,58)(20,41,31,51)(21,46,32,56)(22,39,33,49)(23,44,34,54)(24,37,35,59)(61,88,102,82)(62,93,103,75)(63,86,104,80)(64,91,105,73)(65,96,106,78)(66,89,107,83)(67,94,108,76)(68,87,97,81)(69,92,98,74)(70,85,99,79)(71,90,100,84)(72,95,101,77)(109,111)(110,116)(112,114)(113,119)(115,117)(118,120)>;
G:=Group( (1,110)(2,111)(3,112)(4,113)(5,114)(6,115)(7,116)(8,117)(9,118)(10,119)(11,120)(12,109)(13,106)(14,107)(15,108)(16,97)(17,98)(18,99)(19,100)(20,101)(21,102)(22,103)(23,104)(24,105)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,61)(33,62)(34,63)(35,64)(36,65)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,87)(50,88)(51,89)(52,90)(53,91)(54,92)(55,93)(56,94)(57,95)(58,96)(59,85)(60,86), (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), (1,71,90,84,100)(2,72,91,73,101)(3,61,92,74,102)(4,62,93,75,103)(5,63,94,76,104)(6,64,95,77,105)(7,65,96,78,106)(8,66,85,79,107)(9,67,86,80,108)(10,68,87,81,97)(11,69,88,82,98)(12,70,89,83,99)(13,116,36,58,48)(14,117,25,59,37)(15,118,26,60,38)(16,119,27,49,39)(17,120,28,50,40)(18,109,29,51,41)(19,110,30,52,42)(20,111,31,53,43)(21,112,32,54,44)(22,113,33,55,45)(23,114,34,56,46)(24,115,35,57,47), (2,6)(3,11)(5,9)(8,12)(13,42,36,52)(14,47,25,57)(15,40,26,50)(16,45,27,55)(17,38,28,60)(18,43,29,53)(19,48,30,58)(20,41,31,51)(21,46,32,56)(22,39,33,49)(23,44,34,54)(24,37,35,59)(61,88,102,82)(62,93,103,75)(63,86,104,80)(64,91,105,73)(65,96,106,78)(66,89,107,83)(67,94,108,76)(68,87,97,81)(69,92,98,74)(70,85,99,79)(71,90,100,84)(72,95,101,77)(109,111)(110,116)(112,114)(113,119)(115,117)(118,120) );
G=PermutationGroup([[(1,110),(2,111),(3,112),(4,113),(5,114),(6,115),(7,116),(8,117),(9,118),(10,119),(11,120),(12,109),(13,106),(14,107),(15,108),(16,97),(17,98),(18,99),(19,100),(20,101),(21,102),(22,103),(23,104),(24,105),(25,66),(26,67),(27,68),(28,69),(29,70),(30,71),(31,72),(32,61),(33,62),(34,63),(35,64),(36,65),(37,79),(38,80),(39,81),(40,82),(41,83),(42,84),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,87),(50,88),(51,89),(52,90),(53,91),(54,92),(55,93),(56,94),(57,95),(58,96),(59,85),(60,86)], [(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)], [(1,71,90,84,100),(2,72,91,73,101),(3,61,92,74,102),(4,62,93,75,103),(5,63,94,76,104),(6,64,95,77,105),(7,65,96,78,106),(8,66,85,79,107),(9,67,86,80,108),(10,68,87,81,97),(11,69,88,82,98),(12,70,89,83,99),(13,116,36,58,48),(14,117,25,59,37),(15,118,26,60,38),(16,119,27,49,39),(17,120,28,50,40),(18,109,29,51,41),(19,110,30,52,42),(20,111,31,53,43),(21,112,32,54,44),(22,113,33,55,45),(23,114,34,56,46),(24,115,35,57,47)], [(2,6),(3,11),(5,9),(8,12),(13,42,36,52),(14,47,25,57),(15,40,26,50),(16,45,27,55),(17,38,28,60),(18,43,29,53),(19,48,30,58),(20,41,31,51),(21,46,32,56),(22,39,33,49),(23,44,34,54),(24,37,35,59),(61,88,102,82),(62,93,103,75),(63,86,104,80),(64,91,105,73),(65,96,106,78),(66,89,107,83),(67,94,108,76),(68,87,97,81),(69,92,98,74),(70,85,99,79),(71,90,100,84),(72,95,101,77),(109,111),(110,116),(112,114),(113,119),(115,117),(118,120)]])
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 |
Matrix representation of (C2×C12)⋊6F5 ►in 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] >;
(C2×C12)⋊6F5 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