direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C20.D4, C60.118D4, (C6×D4).7D5, C20.8(C3×D4), (D4×C10).2C6, (D4×C30).7C2, C4.Dic5⋊3C6, C23.(C3×Dic5), (C22×C30).3C4, (C2×C12).213D10, C12.92(C5⋊D4), C15⋊11(C4.D4), (C22×C10).4C12, C22.2(C6×Dic5), (C22×C6).1Dic5, (C2×C60).280C22, C6.23(C23.D5), C30.111(C22⋊C4), (C2×C4).3(C6×D5), C5⋊3(C3×C4.D4), (C2×D4).2(C3×D5), C4.13(C3×C5⋊D4), (C2×C20).16(C2×C6), (C2×C30).185(C2×C4), (C2×C10).49(C2×C12), C2.4(C3×C23.D5), C10.25(C3×C22⋊C4), (C3×C4.Dic5)⋊15C2, (C2×C6).20(C2×Dic5), SmallGroup(480,111)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C20.D4
G = < a,b,c,d | a3=b20=1, c4=b10, d2=b5, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b9, dcd-1=b15c3 >
Subgroups: 224 in 92 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C8, C2×C4, D4, C23, C10, C10, C12, C2×C6, C2×C6, C15, M4(2), C2×D4, C20, C2×C10, C2×C10, C24, C2×C12, C3×D4, C22×C6, C30, C30, C4.D4, C5⋊2C8, C2×C20, C5×D4, C22×C10, C3×M4(2), C6×D4, C60, C2×C30, C2×C30, C4.Dic5, D4×C10, C3×C4.D4, C3×C5⋊2C8, C2×C60, D4×C15, C22×C30, C20.D4, C3×C4.Dic5, D4×C30, C3×C20.D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, D5, C12, C2×C6, C22⋊C4, Dic5, D10, C2×C12, C3×D4, C3×D5, C4.D4, C2×Dic5, C5⋊D4, C3×C22⋊C4, C3×Dic5, C6×D5, C23.D5, C3×C4.D4, C6×Dic5, C3×C5⋊D4, C20.D4, C3×C23.D5, C3×C20.D4
►On 120 points
Generators in S120
(1 56 29)(2 57 30)(3 58 31)(4 59 32)(5 60 33)(6 41 34)(7 42 35)(8 43 36)(9 44 37)(10 45 38)(11 46 39)(12 47 40)(13 48 21)(14 49 22)(15 50 23)(16 51 24)(17 52 25)(18 53 26)(19 54 27)(20 55 28)(61 106 96)(62 107 97)(63 108 98)(64 109 99)(65 110 100)(66 111 81)(67 112 82)(68 113 83)(69 114 84)(70 115 85)(71 116 86)(72 117 87)(73 118 88)(74 119 89)(75 120 90)(76 101 91)(77 102 92)(78 103 93)(79 104 94)(80 105 95)
(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 61 6 76 11 71 16 66)(2 80 7 75 12 70 17 65)(3 79 8 74 13 69 18 64)(4 78 9 73 14 68 19 63)(5 77 10 72 15 67 20 62)(21 84 26 99 31 94 36 89)(22 83 27 98 32 93 37 88)(23 82 28 97 33 92 38 87)(24 81 29 96 34 91 39 86)(25 100 30 95 35 90 40 85)(41 101 46 116 51 111 56 106)(42 120 47 115 52 110 57 105)(43 119 48 114 53 109 58 104)(44 118 49 113 54 108 59 103)(45 117 50 112 55 107 60 102)
(1 71 6 76 11 61 16 66)(2 80 7 65 12 70 17 75)(3 69 8 74 13 79 18 64)(4 78 9 63 14 68 19 73)(5 67 10 72 15 77 20 62)(21 94 26 99 31 84 36 89)(22 83 27 88 32 93 37 98)(23 92 28 97 33 82 38 87)(24 81 29 86 34 91 39 96)(25 90 30 95 35 100 40 85)(41 101 46 106 51 111 56 116)(42 110 47 115 52 120 57 105)(43 119 48 104 53 109 58 114)(44 108 49 113 54 118 59 103)(45 117 50 102 55 107 60 112)
G:=sub<Sym(120)| (1,56,29)(2,57,30)(3,58,31)(4,59,32)(5,60,33)(6,41,34)(7,42,35)(8,43,36)(9,44,37)(10,45,38)(11,46,39)(12,47,40)(13,48,21)(14,49,22)(15,50,23)(16,51,24)(17,52,25)(18,53,26)(19,54,27)(20,55,28)(61,106,96)(62,107,97)(63,108,98)(64,109,99)(65,110,100)(66,111,81)(67,112,82)(68,113,83)(69,114,84)(70,115,85)(71,116,86)(72,117,87)(73,118,88)(74,119,89)(75,120,90)(76,101,91)(77,102,92)(78,103,93)(79,104,94)(80,105,95), (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), (1,61,6,76,11,71,16,66)(2,80,7,75,12,70,17,65)(3,79,8,74,13,69,18,64)(4,78,9,73,14,68,19,63)(5,77,10,72,15,67,20,62)(21,84,26,99,31,94,36,89)(22,83,27,98,32,93,37,88)(23,82,28,97,33,92,38,87)(24,81,29,96,34,91,39,86)(25,100,30,95,35,90,40,85)(41,101,46,116,51,111,56,106)(42,120,47,115,52,110,57,105)(43,119,48,114,53,109,58,104)(44,118,49,113,54,108,59,103)(45,117,50,112,55,107,60,102), (1,71,6,76,11,61,16,66)(2,80,7,65,12,70,17,75)(3,69,8,74,13,79,18,64)(4,78,9,63,14,68,19,73)(5,67,10,72,15,77,20,62)(21,94,26,99,31,84,36,89)(22,83,27,88,32,93,37,98)(23,92,28,97,33,82,38,87)(24,81,29,86,34,91,39,96)(25,90,30,95,35,100,40,85)(41,101,46,106,51,111,56,116)(42,110,47,115,52,120,57,105)(43,119,48,104,53,109,58,114)(44,108,49,113,54,118,59,103)(45,117,50,102,55,107,60,112)>;
G:=Group( (1,56,29)(2,57,30)(3,58,31)(4,59,32)(5,60,33)(6,41,34)(7,42,35)(8,43,36)(9,44,37)(10,45,38)(11,46,39)(12,47,40)(13,48,21)(14,49,22)(15,50,23)(16,51,24)(17,52,25)(18,53,26)(19,54,27)(20,55,28)(61,106,96)(62,107,97)(63,108,98)(64,109,99)(65,110,100)(66,111,81)(67,112,82)(68,113,83)(69,114,84)(70,115,85)(71,116,86)(72,117,87)(73,118,88)(74,119,89)(75,120,90)(76,101,91)(77,102,92)(78,103,93)(79,104,94)(80,105,95), (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), (1,61,6,76,11,71,16,66)(2,80,7,75,12,70,17,65)(3,79,8,74,13,69,18,64)(4,78,9,73,14,68,19,63)(5,77,10,72,15,67,20,62)(21,84,26,99,31,94,36,89)(22,83,27,98,32,93,37,88)(23,82,28,97,33,92,38,87)(24,81,29,96,34,91,39,86)(25,100,30,95,35,90,40,85)(41,101,46,116,51,111,56,106)(42,120,47,115,52,110,57,105)(43,119,48,114,53,109,58,104)(44,118,49,113,54,108,59,103)(45,117,50,112,55,107,60,102), (1,71,6,76,11,61,16,66)(2,80,7,65,12,70,17,75)(3,69,8,74,13,79,18,64)(4,78,9,63,14,68,19,73)(5,67,10,72,15,77,20,62)(21,94,26,99,31,84,36,89)(22,83,27,88,32,93,37,98)(23,92,28,97,33,82,38,87)(24,81,29,86,34,91,39,96)(25,90,30,95,35,100,40,85)(41,101,46,106,51,111,56,116)(42,110,47,115,52,120,57,105)(43,119,48,104,53,109,58,114)(44,108,49,113,54,118,59,103)(45,117,50,102,55,107,60,112) );
G=PermutationGroup([[(1,56,29),(2,57,30),(3,58,31),(4,59,32),(5,60,33),(6,41,34),(7,42,35),(8,43,36),(9,44,37),(10,45,38),(11,46,39),(12,47,40),(13,48,21),(14,49,22),(15,50,23),(16,51,24),(17,52,25),(18,53,26),(19,54,27),(20,55,28),(61,106,96),(62,107,97),(63,108,98),(64,109,99),(65,110,100),(66,111,81),(67,112,82),(68,113,83),(69,114,84),(70,115,85),(71,116,86),(72,117,87),(73,118,88),(74,119,89),(75,120,90),(76,101,91),(77,102,92),(78,103,93),(79,104,94),(80,105,95)], [(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)], [(1,61,6,76,11,71,16,66),(2,80,7,75,12,70,17,65),(3,79,8,74,13,69,18,64),(4,78,9,73,14,68,19,63),(5,77,10,72,15,67,20,62),(21,84,26,99,31,94,36,89),(22,83,27,98,32,93,37,88),(23,82,28,97,33,92,38,87),(24,81,29,96,34,91,39,86),(25,100,30,95,35,90,40,85),(41,101,46,116,51,111,56,106),(42,120,47,115,52,110,57,105),(43,119,48,114,53,109,58,104),(44,118,49,113,54,108,59,103),(45,117,50,112,55,107,60,102)], [(1,71,6,76,11,61,16,66),(2,80,7,65,12,70,17,75),(3,69,8,74,13,79,18,64),(4,78,9,63,14,68,19,73),(5,67,10,72,15,77,20,62),(21,94,26,99,31,84,36,89),(22,83,27,88,32,93,37,98),(23,92,28,97,33,82,38,87),(24,81,29,86,34,91,39,96),(25,90,30,95,35,100,40,85),(41,101,46,106,51,111,56,116),(42,110,47,115,52,120,57,105),(43,119,48,104,53,109,58,114),(44,108,49,113,54,118,59,103),(45,117,50,102,55,107,60,112)]])
93 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 24A | ··· | 24H | 30A | ··· | 30L | 30M | ··· | 30AB | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
93 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | ||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | D4 | D5 | D10 | Dic5 | C3×D4 | C3×D5 | C5⋊D4 | C6×D5 | C3×Dic5 | C3×C5⋊D4 | C4.D4 | C3×C4.D4 | C20.D4 | C3×C20.D4 |
| kernel | C3×C20.D4 | C3×C4.Dic5 | D4×C30 | C20.D4 | C22×C30 | C4.Dic5 | D4×C10 | C22×C10 | C60 | C6×D4 | C2×C12 | C22×C6 | C20 | C2×D4 | C12 | C2×C4 | C23 | C4 | C15 | C5 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 4 | 8 | 16 | 1 | 2 | 4 | 8 |
Matrix representation of C3×C20.D4 ►in GL4(𝔽241) generated by
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 98 | 0 | 0 |
| 143 | 0 | 0 | 0 |
| 0 | 0 | 0 | 91 |
| 0 | 0 | 150 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 240 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 |
G:=sub<GL(4,GF(241))| [15,0,0,0,0,15,0,0,0,0,15,0,0,0,0,15],[0,143,0,0,98,0,0,0,0,0,0,150,0,0,91,0],[0,0,0,1,0,0,1,0,1,0,0,0,0,240,0,0],[0,0,0,240,0,0,1,0,1,0,0,0,0,1,0,0] >;
C3×C20.D4 in GAP, Magma, Sage, TeX
C_3\times C_{20}.D_4 % in TeX
G:=Group("C3xC20.D4"); // GroupNames label
G:=SmallGroup(480,111);
// by ID
G=gap.SmallGroup(480,111);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,850,136,2524,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^20=1,c^4=b^10,d^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^9,d*c*d^-1=b^15*c^3>;
// generators/relations