direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3xD20:4C4, D20:4C12, C60.186D4, C12.69D20, Dic10:4C12, C15:16C4wrC2, (C4xC20):12C6, (C4xC60):14C2, (C4xC12):10D5, C4.6(D5xC12), C42:6(C3xD5), (C3xD20):13C4, C4oD20.1C6, C20.33(C3xD4), C4.17(C3xD20), C12.63(C4xD5), C4.Dic5:1C6, C60.198(C2xC4), C20.37(C2xC12), (C2xC30).152D4, (C3xDic10):13C4, (C2xC12).420D10, C30.80(C22:C4), (C2xC60).516C22, C6.33(D10:C4), C5:3(C3xC4wrC2), (C2xC4).67(C6xD5), (C2xC20).99(C2xC6), (C3xC4oD20).7C2, (C2xC10).27(C3xD4), C22.7(C3xC5:D4), C2.3(C3xD10:C4), (C2xC6).60(C5:D4), C10.12(C3xC22:C4), (C3xC4.Dic5):13C2, SmallGroup(480,83)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3xD20:4C4
G = < a,b,c,d | a3=b20=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b15c >
Subgroups: 288 in 88 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2xC4, C2xC4, D4, Q8, D5, C10, C10, C12, C12, C2xC6, C2xC6, C15, C42, M4(2), C4oD4, Dic5, C20, C20, D10, C2xC10, C24, C2xC12, C2xC12, C3xD4, C3xQ8, C3xD5, C30, C30, C4wrC2, C5:2C8, Dic10, C4xD5, D20, C5:D4, C2xC20, C2xC20, C4xC12, C3xM4(2), C3xC4oD4, C3xDic5, C60, C60, C6xD5, C2xC30, C4.Dic5, C4xC20, C4oD20, C3xC4wrC2, C3xC5:2C8, C3xDic10, D5xC12, C3xD20, C3xC5:D4, C2xC60, C2xC60, D20:4C4, C3xC4.Dic5, C4xC60, C3xC4oD20, C3xD20:4C4
Quotients: C1, C2, C3, C4, C22, C6, C2xC4, D4, D5, C12, C2xC6, C22:C4, D10, C2xC12, C3xD4, C3xD5, C4wrC2, C4xD5, D20, C5:D4, C3xC22:C4, C6xD5, D10:C4, C3xC4wrC2, D5xC12, C3xD20, C3xC5:D4, D20:4C4, C3xD10:C4, C3xD20:4C4
►On 120 points
Generators in S120
(1 51 21)(2 52 22)(3 53 23)(4 54 24)(5 55 25)(6 56 26)(7 57 27)(8 58 28)(9 59 29)(10 60 30)(11 41 31)(12 42 32)(13 43 33)(14 44 34)(15 45 35)(16 46 36)(17 47 37)(18 48 38)(19 49 39)(20 50 40)(61 117 91)(62 118 92)(63 119 93)(64 120 94)(65 101 95)(66 102 96)(67 103 97)(68 104 98)(69 105 99)(70 106 100)(71 107 81)(72 108 82)(73 109 83)(74 110 84)(75 111 85)(76 112 86)(77 113 87)(78 114 88)(79 115 89)(80 116 90)
(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 73)(2 72)(3 71)(4 70)(5 69)(6 68)(7 67)(8 66)(9 65)(10 64)(11 63)(12 62)(13 61)(14 80)(15 79)(16 78)(17 77)(18 76)(19 75)(20 74)(21 83)(22 82)(23 81)(24 100)(25 99)(26 98)(27 97)(28 96)(29 95)(30 94)(31 93)(32 92)(33 91)(34 90)(35 89)(36 88)(37 87)(38 86)(39 85)(40 84)(41 119)(42 118)(43 117)(44 116)(45 115)(46 114)(47 113)(48 112)(49 111)(50 110)(51 109)(52 108)(53 107)(54 106)(55 105)(56 104)(57 103)(58 102)(59 101)(60 120)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(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)
G:=sub<Sym(120)| (1,51,21)(2,52,22)(3,53,23)(4,54,24)(5,55,25)(6,56,26)(7,57,27)(8,58,28)(9,59,29)(10,60,30)(11,41,31)(12,42,32)(13,43,33)(14,44,34)(15,45,35)(16,46,36)(17,47,37)(18,48,38)(19,49,39)(20,50,40)(61,117,91)(62,118,92)(63,119,93)(64,120,94)(65,101,95)(66,102,96)(67,103,97)(68,104,98)(69,105,99)(70,106,100)(71,107,81)(72,108,82)(73,109,83)(74,110,84)(75,111,85)(76,112,86)(77,113,87)(78,114,88)(79,115,89)(80,116,90), (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), (1,73)(2,72)(3,71)(4,70)(5,69)(6,68)(7,67)(8,66)(9,65)(10,64)(11,63)(12,62)(13,61)(14,80)(15,79)(16,78)(17,77)(18,76)(19,75)(20,74)(21,83)(22,82)(23,81)(24,100)(25,99)(26,98)(27,97)(28,96)(29,95)(30,94)(31,93)(32,92)(33,91)(34,90)(35,89)(36,88)(37,87)(38,86)(39,85)(40,84)(41,119)(42,118)(43,117)(44,116)(45,115)(46,114)(47,113)(48,112)(49,111)(50,110)(51,109)(52,108)(53,107)(54,106)(55,105)(56,104)(57,103)(58,102)(59,101)(60,120), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(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)>;
G:=Group( (1,51,21)(2,52,22)(3,53,23)(4,54,24)(5,55,25)(6,56,26)(7,57,27)(8,58,28)(9,59,29)(10,60,30)(11,41,31)(12,42,32)(13,43,33)(14,44,34)(15,45,35)(16,46,36)(17,47,37)(18,48,38)(19,49,39)(20,50,40)(61,117,91)(62,118,92)(63,119,93)(64,120,94)(65,101,95)(66,102,96)(67,103,97)(68,104,98)(69,105,99)(70,106,100)(71,107,81)(72,108,82)(73,109,83)(74,110,84)(75,111,85)(76,112,86)(77,113,87)(78,114,88)(79,115,89)(80,116,90), (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), (1,73)(2,72)(3,71)(4,70)(5,69)(6,68)(7,67)(8,66)(9,65)(10,64)(11,63)(12,62)(13,61)(14,80)(15,79)(16,78)(17,77)(18,76)(19,75)(20,74)(21,83)(22,82)(23,81)(24,100)(25,99)(26,98)(27,97)(28,96)(29,95)(30,94)(31,93)(32,92)(33,91)(34,90)(35,89)(36,88)(37,87)(38,86)(39,85)(40,84)(41,119)(42,118)(43,117)(44,116)(45,115)(46,114)(47,113)(48,112)(49,111)(50,110)(51,109)(52,108)(53,107)(54,106)(55,105)(56,104)(57,103)(58,102)(59,101)(60,120), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(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) );
G=PermutationGroup([[(1,51,21),(2,52,22),(3,53,23),(4,54,24),(5,55,25),(6,56,26),(7,57,27),(8,58,28),(9,59,29),(10,60,30),(11,41,31),(12,42,32),(13,43,33),(14,44,34),(15,45,35),(16,46,36),(17,47,37),(18,48,38),(19,49,39),(20,50,40),(61,117,91),(62,118,92),(63,119,93),(64,120,94),(65,101,95),(66,102,96),(67,103,97),(68,104,98),(69,105,99),(70,106,100),(71,107,81),(72,108,82),(73,109,83),(74,110,84),(75,111,85),(76,112,86),(77,113,87),(78,114,88),(79,115,89),(80,116,90)], [(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)], [(1,73),(2,72),(3,71),(4,70),(5,69),(6,68),(7,67),(8,66),(9,65),(10,64),(11,63),(12,62),(13,61),(14,80),(15,79),(16,78),(17,77),(18,76),(19,75),(20,74),(21,83),(22,82),(23,81),(24,100),(25,99),(26,98),(27,97),(28,96),(29,95),(30,94),(31,93),(32,92),(33,91),(34,90),(35,89),(36,88),(37,87),(38,86),(39,85),(40,84),(41,119),(42,118),(43,117),(44,116),(45,115),(46,114),(47,113),(48,112),(49,111),(50,110),(51,109),(52,108),(53,107),(54,106),(55,105),(56,104),(57,103),(58,102),(59,101),(60,120)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(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)]])
138 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | ··· | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | ··· | 12N | 12O | 12P | 15A | 15B | 15C | 15D | 20A | ··· | 20X | 24A | 24B | 24C | 24D | 30A | ··· | 30L | 60A | ··· | 60AV |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 24 | 24 | 24 | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 20 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 20 | 2 | 2 | 1 | 1 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 2 | ··· | 2 |
138 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 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | D4 | D4 | D5 | D10 | C3xD4 | C3xD4 | C3xD5 | C4wrC2 | C4xD5 | D20 | C5:D4 | C6xD5 | C3xC4wrC2 | D5xC12 | C3xD20 | C3xC5:D4 | D20:4C4 | C3xD20:4C4 |
| kernel | C3xD20:4C4 | C3xC4.Dic5 | C4xC60 | C3xC4oD20 | D20:4C4 | C3xDic10 | C3xD20 | C4.Dic5 | C4xC20 | C4oD20 | Dic10 | D20 | C60 | C2xC30 | C4xC12 | C2xC12 | C20 | C2xC10 | C42 | C15 | C12 | C12 | C2xC6 | C2xC4 | C5 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 16 | 32 |
Matrix representation of C3xD20:4C4 ►in GL3(F241) generated by
| 15 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 40 | 0 |
| 0 | 0 | 235 |
| 240 | 0 | 0 |
| 0 | 0 | 235 |
| 0 | 40 | 0 |
| 240 | 0 | 0 |
| 0 | 240 | 0 |
| 0 | 0 | 177 |
G:=sub<GL(3,GF(241))| [15,0,0,0,1,0,0,0,1],[1,0,0,0,40,0,0,0,235],[240,0,0,0,0,40,0,235,0],[240,0,0,0,240,0,0,0,177] >;
C3xD20:4C4 in GAP, Magma, Sage, TeX
C_3\times D_{20}\rtimes_4C_4 % in TeX
G:=Group("C3xD20:4C4"); // GroupNames label
G:=SmallGroup(480,83);
// by ID
G=gap.SmallGroup(480,83);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,365,92,1683,2524,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^20=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^15*c>;
// generators/relations