direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D4.D10, C60.125D4, C60.201C23, D4⋊D5⋊5C6, (C6×D4)⋊9D5, C4○D20⋊3C6, (D4×C10)⋊2C6, D20⋊6(C2×C6), (D4×C30)⋊9C2, D4.D5⋊5C6, D4.6(C6×D5), C10.49(C6×D4), C20.15(C3×D4), C4.Dic5⋊6C6, C15⋊35(C8⋊C22), Dic10⋊5(C2×C6), (C3×D4).35D10, (C2×C30).164D4, C30.403(C2×D4), (C2×C12).239D10, (C3×D20)⋊36C22, C12.95(C5⋊D4), C20.12(C22×C6), (C2×C60).290C22, (D4×C15).40C22, C12.201(C22×D5), (C3×Dic10)⋊32C22, C5⋊4(C3×C8⋊C22), C4.12(D5×C2×C6), C5⋊2C8⋊3(C2×C6), (C2×D4)⋊2(C3×D5), C2.9(C6×C5⋊D4), (C3×D4⋊D5)⋊13C2, (C5×D4).6(C2×C6), (C2×C4).13(C6×D5), C4.16(C3×C5⋊D4), (C3×C4○D20)⋊13C2, (C2×C20).27(C2×C6), (C3×D4.D5)⋊13C2, (C2×C10).39(C3×D4), C6.130(C2×C5⋊D4), (C3×C5⋊2C8)⋊25C22, (C2×C6).63(C5⋊D4), (C3×C4.Dic5)⋊18C2, C22.10(C3×C5⋊D4), SmallGroup(480,725)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D4.D10
G = < a,b,c,d,e | a3=b4=c2=1, d10=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=b2c, ece-1=b-1c, ede-1=d9 >
Subgroups: 416 in 136 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, C20, D10, C2×C10, C2×C10, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×C6, C3×D5, C30, C30, C8⋊C22, C5⋊2C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4, C5×D4, C22×C10, C3×M4(2), C3×D8, C3×SD16, C6×D4, C3×C4○D4, C3×Dic5, C60, C6×D5, C2×C30, C2×C30, C4.Dic5, D4⋊D5, D4.D5, C4○D20, D4×C10, C3×C8⋊C22, C3×C5⋊2C8, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C2×C60, D4×C15, D4×C15, C22×C30, D4.D10, C3×C4.Dic5, C3×D4⋊D5, C3×D4.D5, C3×C4○D20, D4×C30, C3×D4.D10
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, D10, C3×D4, C22×C6, C3×D5, C8⋊C22, C5⋊D4, C22×D5, C6×D4, C6×D5, C2×C5⋊D4, C3×C8⋊C22, C3×C5⋊D4, D5×C2×C6, D4.D10, C6×C5⋊D4, C3×D4.D10
►On 120 points
Generators in S120
(1 59 33)(2 60 34)(3 41 35)(4 42 36)(5 43 37)(6 44 38)(7 45 39)(8 46 40)(9 47 21)(10 48 22)(11 49 23)(12 50 24)(13 51 25)(14 52 26)(15 53 27)(16 54 28)(17 55 29)(18 56 30)(19 57 31)(20 58 32)(61 104 94)(62 105 95)(63 106 96)(64 107 97)(65 108 98)(66 109 99)(67 110 100)(68 111 81)(69 112 82)(70 113 83)(71 114 84)(72 115 85)(73 116 86)(74 117 87)(75 118 88)(76 119 89)(77 120 90)(78 101 91)(79 102 92)(80 103 93)
(1 16 11 6)(2 17 12 7)(3 18 13 8)(4 19 14 9)(5 20 15 10)(21 36 31 26)(22 37 32 27)(23 38 33 28)(24 39 34 29)(25 40 35 30)(41 56 51 46)(42 57 52 47)(43 58 53 48)(44 59 54 49)(45 60 55 50)(61 66 71 76)(62 67 72 77)(63 68 73 78)(64 69 74 79)(65 70 75 80)(81 86 91 96)(82 87 92 97)(83 88 93 98)(84 89 94 99)(85 90 95 100)(101 106 111 116)(102 107 112 117)(103 108 113 118)(104 109 114 119)(105 110 115 120)
(1 11)(3 13)(5 15)(7 17)(9 19)(21 31)(23 33)(25 35)(27 37)(29 39)(41 51)(43 53)(45 55)(47 57)(49 59)(61 76)(62 67)(63 78)(64 69)(65 80)(66 71)(68 73)(70 75)(72 77)(74 79)(81 86)(82 97)(83 88)(84 99)(85 90)(87 92)(89 94)(91 96)(93 98)(95 100)(101 106)(102 117)(103 108)(104 119)(105 110)(107 112)(109 114)(111 116)(113 118)(115 120)
(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 62 11 72)(2 71 12 61)(3 80 13 70)(4 69 14 79)(5 78 15 68)(6 67 16 77)(7 76 17 66)(8 65 18 75)(9 74 19 64)(10 63 20 73)(21 87 31 97)(22 96 32 86)(23 85 33 95)(24 94 34 84)(25 83 35 93)(26 92 36 82)(27 81 37 91)(28 90 38 100)(29 99 39 89)(30 88 40 98)(41 103 51 113)(42 112 52 102)(43 101 53 111)(44 110 54 120)(45 119 55 109)(46 108 56 118)(47 117 57 107)(48 106 58 116)(49 115 59 105)(50 104 60 114)
G:=sub<Sym(120)| (1,59,33)(2,60,34)(3,41,35)(4,42,36)(5,43,37)(6,44,38)(7,45,39)(8,46,40)(9,47,21)(10,48,22)(11,49,23)(12,50,24)(13,51,25)(14,52,26)(15,53,27)(16,54,28)(17,55,29)(18,56,30)(19,57,31)(20,58,32)(61,104,94)(62,105,95)(63,106,96)(64,107,97)(65,108,98)(66,109,99)(67,110,100)(68,111,81)(69,112,82)(70,113,83)(71,114,84)(72,115,85)(73,116,86)(74,117,87)(75,118,88)(76,119,89)(77,120,90)(78,101,91)(79,102,92)(80,103,93), (1,16,11,6)(2,17,12,7)(3,18,13,8)(4,19,14,9)(5,20,15,10)(21,36,31,26)(22,37,32,27)(23,38,33,28)(24,39,34,29)(25,40,35,30)(41,56,51,46)(42,57,52,47)(43,58,53,48)(44,59,54,49)(45,60,55,50)(61,66,71,76)(62,67,72,77)(63,68,73,78)(64,69,74,79)(65,70,75,80)(81,86,91,96)(82,87,92,97)(83,88,93,98)(84,89,94,99)(85,90,95,100)(101,106,111,116)(102,107,112,117)(103,108,113,118)(104,109,114,119)(105,110,115,120), (1,11)(3,13)(5,15)(7,17)(9,19)(21,31)(23,33)(25,35)(27,37)(29,39)(41,51)(43,53)(45,55)(47,57)(49,59)(61,76)(62,67)(63,78)(64,69)(65,80)(66,71)(68,73)(70,75)(72,77)(74,79)(81,86)(82,97)(83,88)(84,99)(85,90)(87,92)(89,94)(91,96)(93,98)(95,100)(101,106)(102,117)(103,108)(104,119)(105,110)(107,112)(109,114)(111,116)(113,118)(115,120), (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), (1,62,11,72)(2,71,12,61)(3,80,13,70)(4,69,14,79)(5,78,15,68)(6,67,16,77)(7,76,17,66)(8,65,18,75)(9,74,19,64)(10,63,20,73)(21,87,31,97)(22,96,32,86)(23,85,33,95)(24,94,34,84)(25,83,35,93)(26,92,36,82)(27,81,37,91)(28,90,38,100)(29,99,39,89)(30,88,40,98)(41,103,51,113)(42,112,52,102)(43,101,53,111)(44,110,54,120)(45,119,55,109)(46,108,56,118)(47,117,57,107)(48,106,58,116)(49,115,59,105)(50,104,60,114)>;
G:=Group( (1,59,33)(2,60,34)(3,41,35)(4,42,36)(5,43,37)(6,44,38)(7,45,39)(8,46,40)(9,47,21)(10,48,22)(11,49,23)(12,50,24)(13,51,25)(14,52,26)(15,53,27)(16,54,28)(17,55,29)(18,56,30)(19,57,31)(20,58,32)(61,104,94)(62,105,95)(63,106,96)(64,107,97)(65,108,98)(66,109,99)(67,110,100)(68,111,81)(69,112,82)(70,113,83)(71,114,84)(72,115,85)(73,116,86)(74,117,87)(75,118,88)(76,119,89)(77,120,90)(78,101,91)(79,102,92)(80,103,93), (1,16,11,6)(2,17,12,7)(3,18,13,8)(4,19,14,9)(5,20,15,10)(21,36,31,26)(22,37,32,27)(23,38,33,28)(24,39,34,29)(25,40,35,30)(41,56,51,46)(42,57,52,47)(43,58,53,48)(44,59,54,49)(45,60,55,50)(61,66,71,76)(62,67,72,77)(63,68,73,78)(64,69,74,79)(65,70,75,80)(81,86,91,96)(82,87,92,97)(83,88,93,98)(84,89,94,99)(85,90,95,100)(101,106,111,116)(102,107,112,117)(103,108,113,118)(104,109,114,119)(105,110,115,120), (1,11)(3,13)(5,15)(7,17)(9,19)(21,31)(23,33)(25,35)(27,37)(29,39)(41,51)(43,53)(45,55)(47,57)(49,59)(61,76)(62,67)(63,78)(64,69)(65,80)(66,71)(68,73)(70,75)(72,77)(74,79)(81,86)(82,97)(83,88)(84,99)(85,90)(87,92)(89,94)(91,96)(93,98)(95,100)(101,106)(102,117)(103,108)(104,119)(105,110)(107,112)(109,114)(111,116)(113,118)(115,120), (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), (1,62,11,72)(2,71,12,61)(3,80,13,70)(4,69,14,79)(5,78,15,68)(6,67,16,77)(7,76,17,66)(8,65,18,75)(9,74,19,64)(10,63,20,73)(21,87,31,97)(22,96,32,86)(23,85,33,95)(24,94,34,84)(25,83,35,93)(26,92,36,82)(27,81,37,91)(28,90,38,100)(29,99,39,89)(30,88,40,98)(41,103,51,113)(42,112,52,102)(43,101,53,111)(44,110,54,120)(45,119,55,109)(46,108,56,118)(47,117,57,107)(48,106,58,116)(49,115,59,105)(50,104,60,114) );
G=PermutationGroup([[(1,59,33),(2,60,34),(3,41,35),(4,42,36),(5,43,37),(6,44,38),(7,45,39),(8,46,40),(9,47,21),(10,48,22),(11,49,23),(12,50,24),(13,51,25),(14,52,26),(15,53,27),(16,54,28),(17,55,29),(18,56,30),(19,57,31),(20,58,32),(61,104,94),(62,105,95),(63,106,96),(64,107,97),(65,108,98),(66,109,99),(67,110,100),(68,111,81),(69,112,82),(70,113,83),(71,114,84),(72,115,85),(73,116,86),(74,117,87),(75,118,88),(76,119,89),(77,120,90),(78,101,91),(79,102,92),(80,103,93)], [(1,16,11,6),(2,17,12,7),(3,18,13,8),(4,19,14,9),(5,20,15,10),(21,36,31,26),(22,37,32,27),(23,38,33,28),(24,39,34,29),(25,40,35,30),(41,56,51,46),(42,57,52,47),(43,58,53,48),(44,59,54,49),(45,60,55,50),(61,66,71,76),(62,67,72,77),(63,68,73,78),(64,69,74,79),(65,70,75,80),(81,86,91,96),(82,87,92,97),(83,88,93,98),(84,89,94,99),(85,90,95,100),(101,106,111,116),(102,107,112,117),(103,108,113,118),(104,109,114,119),(105,110,115,120)], [(1,11),(3,13),(5,15),(7,17),(9,19),(21,31),(23,33),(25,35),(27,37),(29,39),(41,51),(43,53),(45,55),(47,57),(49,59),(61,76),(62,67),(63,78),(64,69),(65,80),(66,71),(68,73),(70,75),(72,77),(74,79),(81,86),(82,97),(83,88),(84,99),(85,90),(87,92),(89,94),(91,96),(93,98),(95,100),(101,106),(102,117),(103,108),(104,119),(105,110),(107,112),(109,114),(111,116),(113,118),(115,120)], [(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)], [(1,62,11,72),(2,71,12,61),(3,80,13,70),(4,69,14,79),(5,78,15,68),(6,67,16,77),(7,76,17,66),(8,65,18,75),(9,74,19,64),(10,63,20,73),(21,87,31,97),(22,96,32,86),(23,85,33,95),(24,94,34,84),(25,83,35,93),(26,92,36,82),(27,81,37,91),(28,90,38,100),(29,99,39,89),(30,88,40,98),(41,103,51,113),(42,112,52,102),(43,101,53,111),(44,110,54,120),(45,119,55,109),(46,108,56,118),(47,117,57,107),(48,106,58,116),(49,115,59,105),(50,104,60,114)]])
93 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30A | ··· | 30L | 30M | ··· | 30AB | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 4 | 4 | 20 | 1 | 1 | 2 | 2 | 20 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
93 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | D5 | D10 | D10 | C3×D4 | C3×D4 | C3×D5 | C5⋊D4 | C5⋊D4 | C6×D5 | C6×D5 | C3×C5⋊D4 | C3×C5⋊D4 | C8⋊C22 | C3×C8⋊C22 | D4.D10 | C3×D4.D10 |
| kernel | C3×D4.D10 | C3×C4.Dic5 | C3×D4⋊D5 | C3×D4.D5 | C3×C4○D20 | D4×C30 | D4.D10 | C4.Dic5 | D4⋊D5 | D4.D5 | C4○D20 | D4×C10 | C60 | C2×C30 | C6×D4 | C2×C12 | C3×D4 | C20 | C2×C10 | C2×D4 | C12 | C2×C6 | C2×C4 | D4 | C4 | C22 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of C3×D4.D10 ►in GL6(𝔽241)
| 225 | 0 | 0 | 0 | 0 | 0 |
| 0 | 225 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 155 | 0 | 0 |
| 0 | 0 | 213 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 86 |
| 0 | 0 | 0 | 0 | 28 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 144 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 86 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 213 | 240 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 154 | 11 | 0 | 0 |
| 0 | 0 | 26 | 87 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 37 |
| 0 | 0 | 0 | 0 | 197 | 205 |
| 240 | 77 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 37 |
| 0 | 0 | 0 | 0 | 197 | 205 |
| 0 | 0 | 154 | 11 | 0 | 0 |
| 0 | 0 | 26 | 87 | 0 | 0 |
G:=sub<GL(6,GF(241))| [225,0,0,0,0,0,0,225,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,213,0,0,0,0,155,240,0,0,0,0,0,0,240,28,0,0,0,0,86,1],[1,144,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,86,1,0,0,0,0,0,0,1,213,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,154,26,0,0,0,0,11,87,0,0,0,0,0,0,36,197,0,0,0,0,37,205],[240,0,0,0,0,0,77,1,0,0,0,0,0,0,0,0,154,26,0,0,0,0,11,87,0,0,36,197,0,0,0,0,37,205,0,0] >;
C3×D4.D10 in GAP, Magma, Sage, TeX
C_3\times D_4.D_{10} % in TeX
G:=Group("C3xD4.D10"); // GroupNames label
G:=SmallGroup(480,725);
// by ID
G=gap.SmallGroup(480,725);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,344,590,555,2524,648,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^2=1,d^10=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^9>;
// generators/relations