direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C20.46D4, C60.217D4, C12.63D20, (C2×D20).7C6, C4.11(C3×D20), C20.46(C3×D4), C4.Dic5⋊2C6, (C6×D20).18C2, C15⋊8(C4.D4), (C3×M4(2))⋊7D5, M4(2)⋊3(C3×D5), (C5×M4(2))⋊7C6, C22.4(D5×C12), (C2×C12).211D10, (C22×D5).1C12, C12.114(C5⋊D4), C30.87(C22⋊C4), (C15×M4(2))⋊15C2, (C2×C60).276C22, C6.40(D10⋊C4), (D5×C2×C6).2C4, (C2×C4).1(C6×D5), C5⋊2(C3×C4.D4), (C2×C6).38(C4×D5), C4.21(C3×C5⋊D4), (C2×C20).12(C2×C6), (C2×C30).119(C2×C4), (C2×C10).22(C2×C12), C2.9(C3×D10⋊C4), C10.19(C3×C22⋊C4), (C3×C4.Dic5)⋊14C2, SmallGroup(480,101)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C20.46D4
G = < a,b,c,d | a3=b20=d2=1, c4=b10, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=b15c3 >
Subgroups: 416 in 92 conjugacy classes, 38 normal (34 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C8, C2×C4, D4, C23, D5, C10, C10, C12, C2×C6, C2×C6, C15, M4(2), M4(2), C2×D4, C20, D10, C2×C10, C24, C2×C12, C3×D4, C22×C6, C3×D5, C30, C30, C4.D4, C5⋊2C8, C40, D20, C2×C20, C22×D5, C3×M4(2), C3×M4(2), C6×D4, C60, C6×D5, C2×C30, C4.Dic5, C5×M4(2), C2×D20, C3×C4.D4, C3×C5⋊2C8, C120, C3×D20, C2×C60, D5×C2×C6, C20.46D4, C3×C4.Dic5, C15×M4(2), C6×D20, C3×C20.46D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, D5, C12, C2×C6, C22⋊C4, D10, C2×C12, C3×D4, C3×D5, C4.D4, C4×D5, D20, C5⋊D4, C3×C22⋊C4, C6×D5, D10⋊C4, C3×C4.D4, D5×C12, C3×D20, C3×C5⋊D4, C20.46D4, C3×D10⋊C4, C3×C20.46D4
►On 120 points
Generators in S120
(1 48 31)(2 49 32)(3 50 33)(4 51 34)(5 52 35)(6 53 36)(7 54 37)(8 55 38)(9 56 39)(10 57 40)(11 58 21)(12 59 22)(13 60 23)(14 41 24)(15 42 25)(16 43 26)(17 44 27)(18 45 28)(19 46 29)(20 47 30)(61 118 95)(62 119 96)(63 120 97)(64 101 98)(65 102 99)(66 103 100)(67 104 81)(68 105 82)(69 106 83)(70 107 84)(71 108 85)(72 109 86)(73 110 87)(74 111 88)(75 112 89)(76 113 90)(77 114 91)(78 115 92)(79 116 93)(80 117 94)
(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 16 66 11 71 6 76)(2 80 17 65 12 70 7 75)(3 79 18 64 13 69 8 74)(4 78 19 63 14 68 9 73)(5 77 20 62 15 67 10 72)(21 85 36 90 31 95 26 100)(22 84 37 89 32 94 27 99)(23 83 38 88 33 93 28 98)(24 82 39 87 34 92 29 97)(25 81 40 86 35 91 30 96)(41 105 56 110 51 115 46 120)(42 104 57 109 52 114 47 119)(43 103 58 108 53 113 48 118)(44 102 59 107 54 112 49 117)(45 101 60 106 55 111 50 116)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(56 60)(57 59)(61 71)(62 70)(63 69)(64 68)(65 67)(72 80)(73 79)(74 78)(75 77)(81 99)(82 98)(83 97)(84 96)(85 95)(86 94)(87 93)(88 92)(89 91)(101 105)(102 104)(106 120)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)
G:=sub<Sym(120)| (1,48,31)(2,49,32)(3,50,33)(4,51,34)(5,52,35)(6,53,36)(7,54,37)(8,55,38)(9,56,39)(10,57,40)(11,58,21)(12,59,22)(13,60,23)(14,41,24)(15,42,25)(16,43,26)(17,44,27)(18,45,28)(19,46,29)(20,47,30)(61,118,95)(62,119,96)(63,120,97)(64,101,98)(65,102,99)(66,103,100)(67,104,81)(68,105,82)(69,106,83)(70,107,84)(71,108,85)(72,109,86)(73,110,87)(74,111,88)(75,112,89)(76,113,90)(77,114,91)(78,115,92)(79,116,93)(80,117,94), (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), (1,61,16,66,11,71,6,76)(2,80,17,65,12,70,7,75)(3,79,18,64,13,69,8,74)(4,78,19,63,14,68,9,73)(5,77,20,62,15,67,10,72)(21,85,36,90,31,95,26,100)(22,84,37,89,32,94,27,99)(23,83,38,88,33,93,28,98)(24,82,39,87,34,92,29,97)(25,81,40,86,35,91,30,96)(41,105,56,110,51,115,46,120)(42,104,57,109,52,114,47,119)(43,103,58,108,53,113,48,118)(44,102,59,107,54,112,49,117)(45,101,60,106,55,111,50,116), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(56,60)(57,59)(61,71)(62,70)(63,69)(64,68)(65,67)(72,80)(73,79)(74,78)(75,77)(81,99)(82,98)(83,97)(84,96)(85,95)(86,94)(87,93)(88,92)(89,91)(101,105)(102,104)(106,120)(107,119)(108,118)(109,117)(110,116)(111,115)(112,114)>;
G:=Group( (1,48,31)(2,49,32)(3,50,33)(4,51,34)(5,52,35)(6,53,36)(7,54,37)(8,55,38)(9,56,39)(10,57,40)(11,58,21)(12,59,22)(13,60,23)(14,41,24)(15,42,25)(16,43,26)(17,44,27)(18,45,28)(19,46,29)(20,47,30)(61,118,95)(62,119,96)(63,120,97)(64,101,98)(65,102,99)(66,103,100)(67,104,81)(68,105,82)(69,106,83)(70,107,84)(71,108,85)(72,109,86)(73,110,87)(74,111,88)(75,112,89)(76,113,90)(77,114,91)(78,115,92)(79,116,93)(80,117,94), (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), (1,61,16,66,11,71,6,76)(2,80,17,65,12,70,7,75)(3,79,18,64,13,69,8,74)(4,78,19,63,14,68,9,73)(5,77,20,62,15,67,10,72)(21,85,36,90,31,95,26,100)(22,84,37,89,32,94,27,99)(23,83,38,88,33,93,28,98)(24,82,39,87,34,92,29,97)(25,81,40,86,35,91,30,96)(41,105,56,110,51,115,46,120)(42,104,57,109,52,114,47,119)(43,103,58,108,53,113,48,118)(44,102,59,107,54,112,49,117)(45,101,60,106,55,111,50,116), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(56,60)(57,59)(61,71)(62,70)(63,69)(64,68)(65,67)(72,80)(73,79)(74,78)(75,77)(81,99)(82,98)(83,97)(84,96)(85,95)(86,94)(87,93)(88,92)(89,91)(101,105)(102,104)(106,120)(107,119)(108,118)(109,117)(110,116)(111,115)(112,114) );
G=PermutationGroup([[(1,48,31),(2,49,32),(3,50,33),(4,51,34),(5,52,35),(6,53,36),(7,54,37),(8,55,38),(9,56,39),(10,57,40),(11,58,21),(12,59,22),(13,60,23),(14,41,24),(15,42,25),(16,43,26),(17,44,27),(18,45,28),(19,46,29),(20,47,30),(61,118,95),(62,119,96),(63,120,97),(64,101,98),(65,102,99),(66,103,100),(67,104,81),(68,105,82),(69,106,83),(70,107,84),(71,108,85),(72,109,86),(73,110,87),(74,111,88),(75,112,89),(76,113,90),(77,114,91),(78,115,92),(79,116,93),(80,117,94)], [(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)], [(1,61,16,66,11,71,6,76),(2,80,17,65,12,70,7,75),(3,79,18,64,13,69,8,74),(4,78,19,63,14,68,9,73),(5,77,20,62,15,67,10,72),(21,85,36,90,31,95,26,100),(22,84,37,89,32,94,27,99),(23,83,38,88,33,93,28,98),(24,82,39,87,34,92,29,97),(25,81,40,86,35,91,30,96),(41,105,56,110,51,115,46,120),(42,104,57,109,52,114,47,119),(43,103,58,108,53,113,48,118),(44,102,59,107,54,112,49,117),(45,101,60,106,55,111,50,116)], [(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(22,40),(23,39),(24,38),(25,37),(26,36),(27,35),(28,34),(29,33),(30,32),(41,55),(42,54),(43,53),(44,52),(45,51),(46,50),(47,49),(56,60),(57,59),(61,71),(62,70),(63,69),(64,68),(65,67),(72,80),(73,79),(74,78),(75,77),(81,99),(82,98),(83,97),(84,96),(85,95),(86,94),(87,93),(88,92),(89,91),(101,105),(102,104),(106,120),(107,119),(108,118),(109,117),(110,116),(111,115),(112,114)]])
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 | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | 40A | ··· | 40H | 60A | ··· | 60H | 60I | 60J | 60K | 60L | 120A | ··· | 120P |
| 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 | 20 | 20 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 20 | 20 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 20 | 20 | 20 | 20 | 4 | 4 | 20 | 20 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
93 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D4 | D5 | D10 | C3×D4 | C3×D5 | D20 | C5⋊D4 | C4×D5 | C6×D5 | C3×D20 | C3×C5⋊D4 | D5×C12 | C4.D4 | C3×C4.D4 | C20.46D4 | C3×C20.46D4 |
| kernel | C3×C20.46D4 | C3×C4.Dic5 | C15×M4(2) | C6×D20 | C20.46D4 | D5×C2×C6 | C4.Dic5 | C5×M4(2) | C2×D20 | C22×D5 | C60 | C3×M4(2) | C2×C12 | C20 | M4(2) | C12 | C12 | C2×C6 | C2×C4 | C4 | C4 | C22 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of C3×C20.46D4 ►in GL4(𝔽241) generated by
| 225 | 0 | 0 | 0 |
| 0 | 225 | 0 | 0 |
| 0 | 0 | 225 | 0 |
| 0 | 0 | 0 | 225 |
| 85 | 41 | 0 | 0 |
| 200 | 122 | 0 | 0 |
| 0 | 0 | 85 | 41 |
| 0 | 0 | 200 | 122 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 52 | 1 |
| 200 | 156 | 0 | 0 |
| 119 | 41 | 0 | 0 |
| 240 | 0 | 0 | 0 |
| 52 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 52 | 1 |
G:=sub<GL(4,GF(241))| [225,0,0,0,0,225,0,0,0,0,225,0,0,0,0,225],[85,200,0,0,41,122,0,0,0,0,85,200,0,0,41,122],[0,0,200,119,0,0,156,41,240,52,0,0,0,1,0,0],[240,52,0,0,0,1,0,0,0,0,240,52,0,0,0,1] >;
C3×C20.46D4 in GAP, Magma, Sage, TeX
C_3\times C_{20}._{46}D_4 % in TeX
G:=Group("C3xC20.46D4"); // GroupNames label
G:=SmallGroup(480,101);
// by ID
G=gap.SmallGroup(480,101);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,365,92,1683,136,1271,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^20=d^2=1,c^4=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=b^15*c^3>;
// generators/relations