direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3xD4:2Dic5, C60.234D4, C15:22C4wrC2, (C5xD4):5C12, (C5xQ8):8C12, (D4xC15):11C4, (Q8xC15):11C4, Q8:3(C3xDic5), (C4xDic5):2C6, (C3xD4):5Dic5, (C3xQ8):5Dic5, D4:2(C3xDic5), C20.56(C3xD4), (C2xC30).80D4, C4.Dic5:4C6, C4.3(C6xDic5), C20.30(C2xC12), C60.164(C2xC4), (C12xDic5):14C2, (C2xC12).356D10, C12.32(C2xDic5), C12.124(C5:D4), (C2xC60).283C22, C6.27(C23.D5), C30.115(C22:C4), C5:5(C3xC4wrC2), C4oD4.3(C3xD5), (C3xC4oD4).4D5, (C5xC4oD4).5C6, (C2xC4).35(C6xD5), (C2xC10).3(C3xD4), C4.31(C3xC5:D4), (C2xC20).19(C2xC6), (C15xC4oD4).4C2, C22.3(C3xC5:D4), C2.8(C3xC23.D5), (C2xC6).39(C5:D4), C10.29(C3xC22:C4), (C3xC4.Dic5):16C2, SmallGroup(480,115)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3xD4:2Dic5
G = < a,b,c,d,e | a3=b4=d10=1, c2=b2, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b2c, ece-1=b-1c, ede-1=d-1 >
Subgroups: 224 in 88 conjugacy classes, 42 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2xC4, C2xC4, D4, D4, Q8, C10, C10, C12, C12, C2xC6, C2xC6, C15, C42, M4(2), C4oD4, Dic5, C20, C20, C2xC10, C2xC10, C24, C2xC12, C2xC12, C3xD4, C3xD4, C3xQ8, C30, C30, C4wrC2, C5:2C8, C2xDic5, C2xC20, C2xC20, C5xD4, C5xD4, C5xQ8, C4xC12, C3xM4(2), C3xC4oD4, C3xDic5, C60, C60, C2xC30, C2xC30, C4.Dic5, C4xDic5, C5xC4oD4, C3xC4wrC2, C3xC5:2C8, C6xDic5, C2xC60, C2xC60, D4xC15, D4xC15, Q8xC15, D4:2Dic5, C3xC4.Dic5, C12xDic5, C15xC4oD4, C3xD4:2Dic5
Quotients: C1, C2, C3, C4, C22, C6, C2xC4, D4, D5, C12, C2xC6, C22:C4, Dic5, D10, C2xC12, C3xD4, C3xD5, C4wrC2, C2xDic5, C5:D4, C3xC22:C4, C3xDic5, C6xD5, C23.D5, C3xC4wrC2, C6xDic5, C3xC5:D4, D4:2Dic5, C3xC23.D5, C3xD4:2Dic5
►On 120 points
Generators in S120
(1 41 21)(2 42 22)(3 43 23)(4 44 24)(5 45 25)(6 46 26)(7 47 27)(8 48 28)(9 49 29)(10 50 30)(11 51 31)(12 52 32)(13 53 33)(14 54 34)(15 55 35)(16 56 36)(17 57 37)(18 58 38)(19 59 39)(20 60 40)(61 101 81)(62 102 82)(63 103 83)(64 104 84)(65 105 85)(66 106 86)(67 107 87)(68 108 88)(69 109 89)(70 110 90)(71 111 91)(72 112 92)(73 113 93)(74 114 94)(75 115 95)(76 116 96)(77 117 97)(78 118 98)(79 119 99)(80 120 100)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 80)(81 91 86 96)(82 92 87 97)(83 93 88 98)(84 94 89 99)(85 95 90 100)(101 111 106 116)(102 112 107 117)(103 113 108 118)(104 114 109 119)(105 115 110 120)
(1 71 6 76)(2 77 7 72)(3 73 8 78)(4 79 9 74)(5 75 10 80)(11 66 16 61)(12 62 17 67)(13 68 18 63)(14 64 19 69)(15 70 20 65)(21 91 26 96)(22 97 27 92)(23 93 28 98)(24 99 29 94)(25 95 30 100)(31 86 36 81)(32 82 37 87)(33 88 38 83)(34 84 39 89)(35 90 40 85)(41 111 46 116)(42 117 47 112)(43 113 48 118)(44 119 49 114)(45 115 50 120)(51 106 56 101)(52 102 57 107)(53 108 58 103)(54 104 59 109)(55 110 60 105)
(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 6)(2 10)(3 9)(4 8)(5 7)(11 16)(12 20)(13 19)(14 18)(15 17)(21 26)(22 30)(23 29)(24 28)(25 27)(31 36)(32 40)(33 39)(34 38)(35 37)(41 46)(42 50)(43 49)(44 48)(45 47)(51 56)(52 60)(53 59)(54 58)(55 57)(61 71 66 76)(62 80 67 75)(63 79 68 74)(64 78 69 73)(65 77 70 72)(81 91 86 96)(82 100 87 95)(83 99 88 94)(84 98 89 93)(85 97 90 92)(101 111 106 116)(102 120 107 115)(103 119 108 114)(104 118 109 113)(105 117 110 112)
G:=sub<Sym(120)| (1,41,21)(2,42,22)(3,43,23)(4,44,24)(5,45,25)(6,46,26)(7,47,27)(8,48,28)(9,49,29)(10,50,30)(11,51,31)(12,52,32)(13,53,33)(14,54,34)(15,55,35)(16,56,36)(17,57,37)(18,58,38)(19,59,39)(20,60,40)(61,101,81)(62,102,82)(63,103,83)(64,104,84)(65,105,85)(66,106,86)(67,107,87)(68,108,88)(69,109,89)(70,110,90)(71,111,91)(72,112,92)(73,113,93)(74,114,94)(75,115,95)(76,116,96)(77,117,97)(78,118,98)(79,119,99)(80,120,100), (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,71,66,76)(62,72,67,77)(63,73,68,78)(64,74,69,79)(65,75,70,80)(81,91,86,96)(82,92,87,97)(83,93,88,98)(84,94,89,99)(85,95,90,100)(101,111,106,116)(102,112,107,117)(103,113,108,118)(104,114,109,119)(105,115,110,120), (1,71,6,76)(2,77,7,72)(3,73,8,78)(4,79,9,74)(5,75,10,80)(11,66,16,61)(12,62,17,67)(13,68,18,63)(14,64,19,69)(15,70,20,65)(21,91,26,96)(22,97,27,92)(23,93,28,98)(24,99,29,94)(25,95,30,100)(31,86,36,81)(32,82,37,87)(33,88,38,83)(34,84,39,89)(35,90,40,85)(41,111,46,116)(42,117,47,112)(43,113,48,118)(44,119,49,114)(45,115,50,120)(51,106,56,101)(52,102,57,107)(53,108,58,103)(54,104,59,109)(55,110,60,105), (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), (1,6)(2,10)(3,9)(4,8)(5,7)(11,16)(12,20)(13,19)(14,18)(15,17)(21,26)(22,30)(23,29)(24,28)(25,27)(31,36)(32,40)(33,39)(34,38)(35,37)(41,46)(42,50)(43,49)(44,48)(45,47)(51,56)(52,60)(53,59)(54,58)(55,57)(61,71,66,76)(62,80,67,75)(63,79,68,74)(64,78,69,73)(65,77,70,72)(81,91,86,96)(82,100,87,95)(83,99,88,94)(84,98,89,93)(85,97,90,92)(101,111,106,116)(102,120,107,115)(103,119,108,114)(104,118,109,113)(105,117,110,112)>;
G:=Group( (1,41,21)(2,42,22)(3,43,23)(4,44,24)(5,45,25)(6,46,26)(7,47,27)(8,48,28)(9,49,29)(10,50,30)(11,51,31)(12,52,32)(13,53,33)(14,54,34)(15,55,35)(16,56,36)(17,57,37)(18,58,38)(19,59,39)(20,60,40)(61,101,81)(62,102,82)(63,103,83)(64,104,84)(65,105,85)(66,106,86)(67,107,87)(68,108,88)(69,109,89)(70,110,90)(71,111,91)(72,112,92)(73,113,93)(74,114,94)(75,115,95)(76,116,96)(77,117,97)(78,118,98)(79,119,99)(80,120,100), (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,71,66,76)(62,72,67,77)(63,73,68,78)(64,74,69,79)(65,75,70,80)(81,91,86,96)(82,92,87,97)(83,93,88,98)(84,94,89,99)(85,95,90,100)(101,111,106,116)(102,112,107,117)(103,113,108,118)(104,114,109,119)(105,115,110,120), (1,71,6,76)(2,77,7,72)(3,73,8,78)(4,79,9,74)(5,75,10,80)(11,66,16,61)(12,62,17,67)(13,68,18,63)(14,64,19,69)(15,70,20,65)(21,91,26,96)(22,97,27,92)(23,93,28,98)(24,99,29,94)(25,95,30,100)(31,86,36,81)(32,82,37,87)(33,88,38,83)(34,84,39,89)(35,90,40,85)(41,111,46,116)(42,117,47,112)(43,113,48,118)(44,119,49,114)(45,115,50,120)(51,106,56,101)(52,102,57,107)(53,108,58,103)(54,104,59,109)(55,110,60,105), (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), (1,6)(2,10)(3,9)(4,8)(5,7)(11,16)(12,20)(13,19)(14,18)(15,17)(21,26)(22,30)(23,29)(24,28)(25,27)(31,36)(32,40)(33,39)(34,38)(35,37)(41,46)(42,50)(43,49)(44,48)(45,47)(51,56)(52,60)(53,59)(54,58)(55,57)(61,71,66,76)(62,80,67,75)(63,79,68,74)(64,78,69,73)(65,77,70,72)(81,91,86,96)(82,100,87,95)(83,99,88,94)(84,98,89,93)(85,97,90,92)(101,111,106,116)(102,120,107,115)(103,119,108,114)(104,118,109,113)(105,117,110,112) );
G=PermutationGroup([[(1,41,21),(2,42,22),(3,43,23),(4,44,24),(5,45,25),(6,46,26),(7,47,27),(8,48,28),(9,49,29),(10,50,30),(11,51,31),(12,52,32),(13,53,33),(14,54,34),(15,55,35),(16,56,36),(17,57,37),(18,58,38),(19,59,39),(20,60,40),(61,101,81),(62,102,82),(63,103,83),(64,104,84),(65,105,85),(66,106,86),(67,107,87),(68,108,88),(69,109,89),(70,110,90),(71,111,91),(72,112,92),(73,113,93),(74,114,94),(75,115,95),(76,116,96),(77,117,97),(78,118,98),(79,119,99),(80,120,100)], [(1,16,6,11),(2,17,7,12),(3,18,8,13),(4,19,9,14),(5,20,10,15),(21,36,26,31),(22,37,27,32),(23,38,28,33),(24,39,29,34),(25,40,30,35),(41,56,46,51),(42,57,47,52),(43,58,48,53),(44,59,49,54),(45,60,50,55),(61,71,66,76),(62,72,67,77),(63,73,68,78),(64,74,69,79),(65,75,70,80),(81,91,86,96),(82,92,87,97),(83,93,88,98),(84,94,89,99),(85,95,90,100),(101,111,106,116),(102,112,107,117),(103,113,108,118),(104,114,109,119),(105,115,110,120)], [(1,71,6,76),(2,77,7,72),(3,73,8,78),(4,79,9,74),(5,75,10,80),(11,66,16,61),(12,62,17,67),(13,68,18,63),(14,64,19,69),(15,70,20,65),(21,91,26,96),(22,97,27,92),(23,93,28,98),(24,99,29,94),(25,95,30,100),(31,86,36,81),(32,82,37,87),(33,88,38,83),(34,84,39,89),(35,90,40,85),(41,111,46,116),(42,117,47,112),(43,113,48,118),(44,119,49,114),(45,115,50,120),(51,106,56,101),(52,102,57,107),(53,108,58,103),(54,104,59,109),(55,110,60,105)], [(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)], [(1,6),(2,10),(3,9),(4,8),(5,7),(11,16),(12,20),(13,19),(14,18),(15,17),(21,26),(22,30),(23,29),(24,28),(25,27),(31,36),(32,40),(33,39),(34,38),(35,37),(41,46),(42,50),(43,49),(44,48),(45,47),(51,56),(52,60),(53,59),(54,58),(55,57),(61,71,66,76),(62,80,67,75),(63,79,68,74),(64,78,69,73),(65,77,70,72),(81,91,86,96),(82,100,87,95),(83,99,88,94),(84,98,89,93),(85,97,90,92),(101,111,106,116),(102,120,107,115),(103,119,108,114),(104,118,109,113),(105,117,110,112)]])
102 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 10A | 10B | 10C | ··· | 10H | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | ··· | 12P | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | ··· | 20J | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 30E | ··· | 30P | 60A | ··· | 60H | 60I | ··· | 60T |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 4 | 10 | 10 | 10 | 10 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
102 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 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | ||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | D4 | D4 | D5 | D10 | Dic5 | Dic5 | C3xD4 | C3xD4 | C3xD5 | C4wrC2 | C5:D4 | C5:D4 | C6xD5 | C3xDic5 | C3xDic5 | C3xC4wrC2 | C3xC5:D4 | C3xC5:D4 | D4:2Dic5 | C3xD4:2Dic5 |
| kernel | C3xD4:2Dic5 | C3xC4.Dic5 | C12xDic5 | C15xC4oD4 | D4:2Dic5 | D4xC15 | Q8xC15 | C4.Dic5 | C4xDic5 | C5xC4oD4 | C5xD4 | C5xQ8 | C60 | C2xC30 | C3xC4oD4 | C2xC12 | C3xD4 | C3xQ8 | C20 | C2xC10 | C4oD4 | C15 | C12 | C2xC6 | C2xC4 | D4 | Q8 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 4 | 8 |
Matrix representation of C3xD4:2Dic5 ►in GL6(F241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 0 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 64 | 0 | 0 | 0 | 0 | 0 |
| 177 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 64 | 128 | 0 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 240 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 51 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 89 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 239 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 51 | 190 |
| 0 | 0 | 0 | 0 | 240 | 190 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[64,177,0,0,0,0,0,177,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[64,0,0,0,0,0,128,177,0,0,0,0,0,0,1,1,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,240,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,240,51],[240,89,0,0,0,0,0,177,0,0,0,0,0,0,1,0,0,0,0,0,239,240,0,0,0,0,0,0,51,240,0,0,0,0,190,190] >;
C3xD4:2Dic5 in GAP, Magma, Sage, TeX
C_3\times D_4\rtimes_2{\rm Dic}_5 % in TeX
G:=Group("C3xD4:2Dic5"); // GroupNames label
G:=SmallGroup(480,115);
// by ID
G=gap.SmallGroup(480,115);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,136,2524,1271,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=d^10=1,c^2=b^2,e^2=d^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^2*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^-1>;
// generators/relations