metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C30)⋊5D4, (C2×C6)⋊8D20, C15⋊5C22≀C2, (C6×D5)⋊17D4, C6.162(D4×D5), C6.69(C2×D20), (C23×D5)⋊5S3, D10⋊8(C3⋊D4), C3⋊5(C22⋊D20), C5⋊1(C24⋊4S3), (C2×Dic3)⋊4D10, C30.244(C2×D4), C23.44(S3×D5), C6.D4⋊13D5, C22⋊3(C3⋊D20), (C22×D5).95D6, (C22×C10).58D6, (C22×C6).94D10, D10⋊Dic3⋊34C2, (C2×C30).206C23, (C10×Dic3)⋊8C22, (C22×D15)⋊6C22, (C2×Dic15)⋊12C22, (C22×C30).68C22, (D5×C22×C6)⋊2C2, C2.42(D5×C3⋊D4), (C2×C3⋊D20)⋊13C2, (C2×C15⋊7D4)⋊16C2, (C2×C10)⋊6(C3⋊D4), C10.66(C2×C3⋊D4), C2.24(C2×C3⋊D20), C22.235(C2×S3×D5), (D5×C2×C6).113C22, (C5×C6.D4)⋊15C2, (C2×C6).218(C22×D5), (C2×C10).218(C22×S3), SmallGroup(480,640)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C6)⋊8D20
G = < a,b,c,d | a2=b6=c20=d2=1, ab=ba, cac-1=dad=ab3, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 1500 in 260 conjugacy classes, 60 normal (24 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, C10, Dic3, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C22×C6, C3×D5, D15, C30, C30, C30, C22≀C2, D20, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×D5, C22×C10, C6.D4, C6.D4, C2×C3⋊D4, C23×C6, C5×Dic3, Dic15, C6×D5, C6×D5, D30, C2×C30, C2×C30, C2×C30, D10⋊C4, C5×C22⋊C4, C2×D20, C2×C5⋊D4, C23×D5, C24⋊4S3, C3⋊D20, C10×Dic3, C2×Dic15, C15⋊7D4, D5×C2×C6, D5×C2×C6, C22×D15, C22×C30, C22⋊D20, D10⋊Dic3, C5×C6.D4, C2×C3⋊D20, C2×C15⋊7D4, D5×C22×C6, (C2×C6)⋊8D20
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C22≀C2, D20, C22×D5, C2×C3⋊D4, S3×D5, C2×D20, D4×D5, C24⋊4S3, C3⋊D20, C2×S3×D5, C22⋊D20, C2×C3⋊D20, D5×C3⋊D4, (C2×C6)⋊8D20
►On 120 points
Generators in S120
(1 11)(2 45)(3 13)(4 47)(5 15)(6 49)(7 17)(8 51)(9 19)(10 53)(12 55)(14 57)(16 59)(18 41)(20 43)(21 31)(22 109)(23 33)(24 111)(25 35)(26 113)(27 37)(28 115)(29 39)(30 117)(32 119)(34 101)(36 103)(38 105)(40 107)(42 52)(44 54)(46 56)(48 58)(50 60)(61 71)(62 100)(63 73)(64 82)(65 75)(66 84)(67 77)(68 86)(69 79)(70 88)(72 90)(74 92)(76 94)(78 96)(80 98)(81 91)(83 93)(85 95)(87 97)(89 99)(102 112)(104 114)(106 116)(108 118)(110 120)
(1 75 106 54 83 29)(2 30 84 55 107 76)(3 77 108 56 85 31)(4 32 86 57 109 78)(5 79 110 58 87 33)(6 34 88 59 111 80)(7 61 112 60 89 35)(8 36 90 41 113 62)(9 63 114 42 91 37)(10 38 92 43 115 64)(11 65 116 44 93 39)(12 40 94 45 117 66)(13 67 118 46 95 21)(14 22 96 47 119 68)(15 69 120 48 97 23)(16 24 98 49 101 70)(17 71 102 50 99 25)(18 26 100 51 103 72)(19 73 104 52 81 27)(20 28 82 53 105 74)
(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 53)(2 52)(3 51)(4 50)(5 49)(6 48)(7 47)(8 46)(9 45)(10 44)(11 43)(12 42)(13 41)(14 60)(15 59)(16 58)(17 57)(18 56)(19 55)(20 54)(21 113)(22 112)(23 111)(24 110)(25 109)(26 108)(27 107)(28 106)(29 105)(30 104)(31 103)(32 102)(33 101)(34 120)(35 119)(36 118)(37 117)(38 116)(39 115)(40 114)(61 96)(62 95)(63 94)(64 93)(65 92)(66 91)(67 90)(68 89)(69 88)(70 87)(71 86)(72 85)(73 84)(74 83)(75 82)(76 81)(77 100)(78 99)(79 98)(80 97)
G:=sub<Sym(120)| (1,11)(2,45)(3,13)(4,47)(5,15)(6,49)(7,17)(8,51)(9,19)(10,53)(12,55)(14,57)(16,59)(18,41)(20,43)(21,31)(22,109)(23,33)(24,111)(25,35)(26,113)(27,37)(28,115)(29,39)(30,117)(32,119)(34,101)(36,103)(38,105)(40,107)(42,52)(44,54)(46,56)(48,58)(50,60)(61,71)(62,100)(63,73)(64,82)(65,75)(66,84)(67,77)(68,86)(69,79)(70,88)(72,90)(74,92)(76,94)(78,96)(80,98)(81,91)(83,93)(85,95)(87,97)(89,99)(102,112)(104,114)(106,116)(108,118)(110,120), (1,75,106,54,83,29)(2,30,84,55,107,76)(3,77,108,56,85,31)(4,32,86,57,109,78)(5,79,110,58,87,33)(6,34,88,59,111,80)(7,61,112,60,89,35)(8,36,90,41,113,62)(9,63,114,42,91,37)(10,38,92,43,115,64)(11,65,116,44,93,39)(12,40,94,45,117,66)(13,67,118,46,95,21)(14,22,96,47,119,68)(15,69,120,48,97,23)(16,24,98,49,101,70)(17,71,102,50,99,25)(18,26,100,51,103,72)(19,73,104,52,81,27)(20,28,82,53,105,74), (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), (1,53)(2,52)(3,51)(4,50)(5,49)(6,48)(7,47)(8,46)(9,45)(10,44)(11,43)(12,42)(13,41)(14,60)(15,59)(16,58)(17,57)(18,56)(19,55)(20,54)(21,113)(22,112)(23,111)(24,110)(25,109)(26,108)(27,107)(28,106)(29,105)(30,104)(31,103)(32,102)(33,101)(34,120)(35,119)(36,118)(37,117)(38,116)(39,115)(40,114)(61,96)(62,95)(63,94)(64,93)(65,92)(66,91)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,84)(74,83)(75,82)(76,81)(77,100)(78,99)(79,98)(80,97)>;
G:=Group( (1,11)(2,45)(3,13)(4,47)(5,15)(6,49)(7,17)(8,51)(9,19)(10,53)(12,55)(14,57)(16,59)(18,41)(20,43)(21,31)(22,109)(23,33)(24,111)(25,35)(26,113)(27,37)(28,115)(29,39)(30,117)(32,119)(34,101)(36,103)(38,105)(40,107)(42,52)(44,54)(46,56)(48,58)(50,60)(61,71)(62,100)(63,73)(64,82)(65,75)(66,84)(67,77)(68,86)(69,79)(70,88)(72,90)(74,92)(76,94)(78,96)(80,98)(81,91)(83,93)(85,95)(87,97)(89,99)(102,112)(104,114)(106,116)(108,118)(110,120), (1,75,106,54,83,29)(2,30,84,55,107,76)(3,77,108,56,85,31)(4,32,86,57,109,78)(5,79,110,58,87,33)(6,34,88,59,111,80)(7,61,112,60,89,35)(8,36,90,41,113,62)(9,63,114,42,91,37)(10,38,92,43,115,64)(11,65,116,44,93,39)(12,40,94,45,117,66)(13,67,118,46,95,21)(14,22,96,47,119,68)(15,69,120,48,97,23)(16,24,98,49,101,70)(17,71,102,50,99,25)(18,26,100,51,103,72)(19,73,104,52,81,27)(20,28,82,53,105,74), (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), (1,53)(2,52)(3,51)(4,50)(5,49)(6,48)(7,47)(8,46)(9,45)(10,44)(11,43)(12,42)(13,41)(14,60)(15,59)(16,58)(17,57)(18,56)(19,55)(20,54)(21,113)(22,112)(23,111)(24,110)(25,109)(26,108)(27,107)(28,106)(29,105)(30,104)(31,103)(32,102)(33,101)(34,120)(35,119)(36,118)(37,117)(38,116)(39,115)(40,114)(61,96)(62,95)(63,94)(64,93)(65,92)(66,91)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,84)(74,83)(75,82)(76,81)(77,100)(78,99)(79,98)(80,97) );
G=PermutationGroup([[(1,11),(2,45),(3,13),(4,47),(5,15),(6,49),(7,17),(8,51),(9,19),(10,53),(12,55),(14,57),(16,59),(18,41),(20,43),(21,31),(22,109),(23,33),(24,111),(25,35),(26,113),(27,37),(28,115),(29,39),(30,117),(32,119),(34,101),(36,103),(38,105),(40,107),(42,52),(44,54),(46,56),(48,58),(50,60),(61,71),(62,100),(63,73),(64,82),(65,75),(66,84),(67,77),(68,86),(69,79),(70,88),(72,90),(74,92),(76,94),(78,96),(80,98),(81,91),(83,93),(85,95),(87,97),(89,99),(102,112),(104,114),(106,116),(108,118),(110,120)], [(1,75,106,54,83,29),(2,30,84,55,107,76),(3,77,108,56,85,31),(4,32,86,57,109,78),(5,79,110,58,87,33),(6,34,88,59,111,80),(7,61,112,60,89,35),(8,36,90,41,113,62),(9,63,114,42,91,37),(10,38,92,43,115,64),(11,65,116,44,93,39),(12,40,94,45,117,66),(13,67,118,46,95,21),(14,22,96,47,119,68),(15,69,120,48,97,23),(16,24,98,49,101,70),(17,71,102,50,99,25),(18,26,100,51,103,72),(19,73,104,52,81,27),(20,28,82,53,105,74)], [(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)], [(1,53),(2,52),(3,51),(4,50),(5,49),(6,48),(7,47),(8,46),(9,45),(10,44),(11,43),(12,42),(13,41),(14,60),(15,59),(16,58),(17,57),(18,56),(19,55),(20,54),(21,113),(22,112),(23,111),(24,110),(25,109),(26,108),(27,107),(28,106),(29,105),(30,104),(31,103),(32,102),(33,101),(34,120),(35,119),(36,118),(37,117),(38,116),(39,115),(40,114),(61,96),(62,95),(63,94),(64,93),(65,92),(66,91),(67,90),(68,89),(69,88),(70,87),(71,86),(72,85),(73,84),(74,83),(75,82),(76,81),(77,100),(78,99),(79,98),(80,97)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 5A | 5B | 6A | ··· | 6G | 6H | ··· | 6O | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 60 | 2 | 12 | 12 | 60 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D10 | D10 | C3⋊D4 | C3⋊D4 | D20 | S3×D5 | D4×D5 | C3⋊D20 | C2×S3×D5 | D5×C3⋊D4 |
| kernel | (C2×C6)⋊8D20 | D10⋊Dic3 | C5×C6.D4 | C2×C3⋊D20 | C2×C15⋊7D4 | D5×C22×C6 | C23×D5 | C6×D5 | C2×C30 | C6.D4 | C22×D5 | C22×C10 | C2×Dic3 | C22×C6 | D10 | C2×C10 | C2×C6 | C23 | C6 | C22 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 4 | 2 | 8 | 4 | 8 | 2 | 4 | 4 | 2 | 8 |
Matrix representation of (C2×C6)⋊8D20 ►in GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 29 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 14 | 5 |
| 0 | 0 | 0 | 48 |
| 7 | 32 | 0 | 0 |
| 29 | 2 | 0 | 0 |
| 0 | 0 | 55 | 3 |
| 0 | 0 | 8 | 6 |
| 7 | 32 | 0 | 0 |
| 29 | 54 | 0 | 0 |
| 0 | 0 | 55 | 49 |
| 0 | 0 | 8 | 6 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,60,0,0,0,29,1],[1,0,0,0,0,1,0,0,0,0,14,0,0,0,5,48],[7,29,0,0,32,2,0,0,0,0,55,8,0,0,3,6],[7,29,0,0,32,54,0,0,0,0,55,8,0,0,49,6] >;
(C2×C6)⋊8D20 in GAP, Magma, Sage, TeX
(C_2\times C_6)\rtimes_8D_{20} % in TeX
G:=Group("(C2xC6):8D20"); // GroupNames label
G:=SmallGroup(480,640);
// by ID
G=gap.SmallGroup(480,640);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,64,219,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^20=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a*b^3,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations