direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D20⋊7C4, D20⋊7C12, C60.232D4, Dic10⋊7C12, C15⋊19C4≀C2, C4.3(D5×C12), (C3×D20)⋊16C4, (C4×Dic5)⋊1C6, C4○D20.2C6, (C2×C30).78D4, C20.54(C3×D4), (C2×C6).25D20, C12.46(C4×D5), C60.161(C2×C4), C20.27(C2×C12), (C3×M4(2))⋊8D5, (C5×M4(2))⋊8C6, M4(2)⋊4(C3×D5), C22.3(C3×D20), (C12×Dic5)⋊13C2, (C3×Dic10)⋊16C4, (C2×C12).353D10, C12.122(C5⋊D4), C30.89(C22⋊C4), (C15×M4(2))⋊16C2, (C2×C60).278C22, C6.42(D10⋊C4), C5⋊4(C3×C4≀C2), (C2×C10).1(C3×D4), (C2×C4).32(C6×D5), C4.29(C3×C5⋊D4), (C2×C20).14(C2×C6), (C3×C4○D20).8C2, C10.21(C3×C22⋊C4), C2.11(C3×D10⋊C4), SmallGroup(480,103)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D20⋊7C4
G = < a,b,c,d | a3=b20=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b9, dcd-1=b3c >
Subgroups: 320 in 88 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C42, M4(2), C4○D4, Dic5, C20, D10, C2×C10, C24, C2×C12, C2×C12, C3×D4, C3×Q8, C3×D5, C30, C30, C4≀C2, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C4×C12, C3×M4(2), C3×C4○D4, C3×Dic5, C60, C6×D5, C2×C30, C4×Dic5, C5×M4(2), C4○D20, C3×C4≀C2, C120, C3×Dic10, D5×C12, C3×D20, C6×Dic5, C3×C5⋊D4, C2×C60, D20⋊7C4, C12×Dic5, C15×M4(2), C3×C4○D20, C3×D20⋊7C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, D5, C12, C2×C6, C22⋊C4, D10, C2×C12, C3×D4, C3×D5, C4≀C2, C4×D5, D20, C5⋊D4, C3×C22⋊C4, C6×D5, D10⋊C4, C3×C4≀C2, D5×C12, C3×D20, C3×C5⋊D4, D20⋊7C4, C3×D10⋊C4, C3×D20⋊7C4
►On 120 points
Generators in S120
(1 56 31)(2 57 32)(3 58 33)(4 59 34)(5 60 35)(6 41 36)(7 42 37)(8 43 38)(9 44 39)(10 45 40)(11 46 21)(12 47 22)(13 48 23)(14 49 24)(15 50 25)(16 51 26)(17 52 27)(18 53 28)(19 54 29)(20 55 30)(61 101 91)(62 102 92)(63 103 93)(64 104 94)(65 105 95)(66 106 96)(67 107 97)(68 108 98)(69 109 99)(70 110 100)(71 111 81)(72 112 82)(73 113 83)(74 114 84)(75 115 85)(76 116 86)(77 117 87)(78 118 88)(79 119 89)(80 120 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 70)(2 69)(3 68)(4 67)(5 66)(6 65)(7 64)(8 63)(9 62)(10 61)(11 80)(12 79)(13 78)(14 77)(15 76)(16 75)(17 74)(18 73)(19 72)(20 71)(21 90)(22 89)(23 88)(24 87)(25 86)(26 85)(27 84)(28 83)(29 82)(30 81)(31 100)(32 99)(33 98)(34 97)(35 96)(36 95)(37 94)(38 93)(39 92)(40 91)(41 105)(42 104)(43 103)(44 102)(45 101)(46 120)(47 119)(48 118)(49 117)(50 116)(51 115)(52 114)(53 113)(54 112)(55 111)(56 110)(57 109)(58 108)(59 107)(60 106)
(1 16 11 6)(2 5 12 15)(3 14 13 4)(7 10 17 20)(8 19 18 9)(21 36 31 26)(22 25 32 35)(23 34 33 24)(27 30 37 40)(28 39 38 29)(41 56 51 46)(42 45 52 55)(43 54 53 44)(47 50 57 60)(48 59 58 49)(62 70)(63 79)(64 68)(65 77)(67 75)(69 73)(72 80)(74 78)(82 90)(83 99)(84 88)(85 97)(87 95)(89 93)(92 100)(94 98)(102 110)(103 119)(104 108)(105 117)(107 115)(109 113)(112 120)(114 118)
G:=sub<Sym(120)| (1,56,31)(2,57,32)(3,58,33)(4,59,34)(5,60,35)(6,41,36)(7,42,37)(8,43,38)(9,44,39)(10,45,40)(11,46,21)(12,47,22)(13,48,23)(14,49,24)(15,50,25)(16,51,26)(17,52,27)(18,53,28)(19,54,29)(20,55,30)(61,101,91)(62,102,92)(63,103,93)(64,104,94)(65,105,95)(66,106,96)(67,107,97)(68,108,98)(69,109,99)(70,110,100)(71,111,81)(72,112,82)(73,113,83)(74,114,84)(75,115,85)(76,116,86)(77,117,87)(78,118,88)(79,119,89)(80,120,90), (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), (1,70)(2,69)(3,68)(4,67)(5,66)(6,65)(7,64)(8,63)(9,62)(10,61)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,90)(22,89)(23,88)(24,87)(25,86)(26,85)(27,84)(28,83)(29,82)(30,81)(31,100)(32,99)(33,98)(34,97)(35,96)(36,95)(37,94)(38,93)(39,92)(40,91)(41,105)(42,104)(43,103)(44,102)(45,101)(46,120)(47,119)(48,118)(49,117)(50,116)(51,115)(52,114)(53,113)(54,112)(55,111)(56,110)(57,109)(58,108)(59,107)(60,106), (1,16,11,6)(2,5,12,15)(3,14,13,4)(7,10,17,20)(8,19,18,9)(21,36,31,26)(22,25,32,35)(23,34,33,24)(27,30,37,40)(28,39,38,29)(41,56,51,46)(42,45,52,55)(43,54,53,44)(47,50,57,60)(48,59,58,49)(62,70)(63,79)(64,68)(65,77)(67,75)(69,73)(72,80)(74,78)(82,90)(83,99)(84,88)(85,97)(87,95)(89,93)(92,100)(94,98)(102,110)(103,119)(104,108)(105,117)(107,115)(109,113)(112,120)(114,118)>;
G:=Group( (1,56,31)(2,57,32)(3,58,33)(4,59,34)(5,60,35)(6,41,36)(7,42,37)(8,43,38)(9,44,39)(10,45,40)(11,46,21)(12,47,22)(13,48,23)(14,49,24)(15,50,25)(16,51,26)(17,52,27)(18,53,28)(19,54,29)(20,55,30)(61,101,91)(62,102,92)(63,103,93)(64,104,94)(65,105,95)(66,106,96)(67,107,97)(68,108,98)(69,109,99)(70,110,100)(71,111,81)(72,112,82)(73,113,83)(74,114,84)(75,115,85)(76,116,86)(77,117,87)(78,118,88)(79,119,89)(80,120,90), (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), (1,70)(2,69)(3,68)(4,67)(5,66)(6,65)(7,64)(8,63)(9,62)(10,61)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,90)(22,89)(23,88)(24,87)(25,86)(26,85)(27,84)(28,83)(29,82)(30,81)(31,100)(32,99)(33,98)(34,97)(35,96)(36,95)(37,94)(38,93)(39,92)(40,91)(41,105)(42,104)(43,103)(44,102)(45,101)(46,120)(47,119)(48,118)(49,117)(50,116)(51,115)(52,114)(53,113)(54,112)(55,111)(56,110)(57,109)(58,108)(59,107)(60,106), (1,16,11,6)(2,5,12,15)(3,14,13,4)(7,10,17,20)(8,19,18,9)(21,36,31,26)(22,25,32,35)(23,34,33,24)(27,30,37,40)(28,39,38,29)(41,56,51,46)(42,45,52,55)(43,54,53,44)(47,50,57,60)(48,59,58,49)(62,70)(63,79)(64,68)(65,77)(67,75)(69,73)(72,80)(74,78)(82,90)(83,99)(84,88)(85,97)(87,95)(89,93)(92,100)(94,98)(102,110)(103,119)(104,108)(105,117)(107,115)(109,113)(112,120)(114,118) );
G=PermutationGroup([[(1,56,31),(2,57,32),(3,58,33),(4,59,34),(5,60,35),(6,41,36),(7,42,37),(8,43,38),(9,44,39),(10,45,40),(11,46,21),(12,47,22),(13,48,23),(14,49,24),(15,50,25),(16,51,26),(17,52,27),(18,53,28),(19,54,29),(20,55,30),(61,101,91),(62,102,92),(63,103,93),(64,104,94),(65,105,95),(66,106,96),(67,107,97),(68,108,98),(69,109,99),(70,110,100),(71,111,81),(72,112,82),(73,113,83),(74,114,84),(75,115,85),(76,116,86),(77,117,87),(78,118,88),(79,119,89),(80,120,90)], [(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)], [(1,70),(2,69),(3,68),(4,67),(5,66),(6,65),(7,64),(8,63),(9,62),(10,61),(11,80),(12,79),(13,78),(14,77),(15,76),(16,75),(17,74),(18,73),(19,72),(20,71),(21,90),(22,89),(23,88),(24,87),(25,86),(26,85),(27,84),(28,83),(29,82),(30,81),(31,100),(32,99),(33,98),(34,97),(35,96),(36,95),(37,94),(38,93),(39,92),(40,91),(41,105),(42,104),(43,103),(44,102),(45,101),(46,120),(47,119),(48,118),(49,117),(50,116),(51,115),(52,114),(53,113),(54,112),(55,111),(56,110),(57,109),(58,108),(59,107),(60,106)], [(1,16,11,6),(2,5,12,15),(3,14,13,4),(7,10,17,20),(8,19,18,9),(21,36,31,26),(22,25,32,35),(23,34,33,24),(27,30,37,40),(28,39,38,29),(41,56,51,46),(42,45,52,55),(43,54,53,44),(47,50,57,60),(48,59,58,49),(62,70),(63,79),(64,68),(65,77),(67,75),(69,73),(72,80),(74,78),(82,90),(83,99),(84,88),(85,97),(87,95),(89,93),(92,100),(94,98),(102,110),(103,119),(104,108),(105,117),(107,115),(109,113),(112,120),(114,118)]])
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 | 10D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | ··· | 12N | 12O | 12P | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | 40A | ··· | 40H | 60A | ··· | 60H | 60I | 60J | 60K | 60L | 120A | ··· | 120P |
| 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 | 30 | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 20 | 1 | 1 | 1 | 1 | 2 | 10 | 10 | 10 | 10 | 20 | 2 | 2 | 1 | 1 | 2 | 2 | 20 | 20 | 4 | 4 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 10 | ··· | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 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 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | D4 | D4 | D5 | D10 | C3×D4 | C3×D4 | C3×D5 | C4≀C2 | C4×D5 | C5⋊D4 | D20 | C6×D5 | C3×C4≀C2 | D5×C12 | C3×C5⋊D4 | C3×D20 | D20⋊7C4 | C3×D20⋊7C4 |
| kernel | C3×D20⋊7C4 | C12×Dic5 | C15×M4(2) | C3×C4○D20 | D20⋊7C4 | C3×Dic10 | C3×D20 | C4×Dic5 | C5×M4(2) | C4○D20 | Dic10 | D20 | C60 | C2×C30 | C3×M4(2) | C2×C12 | C20 | C2×C10 | M4(2) | C15 | C12 | C12 | C2×C6 | C2×C4 | 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 | 4 | 8 |
Matrix representation of C3×D20⋊7C4 ►in GL4(𝔽241) generated by
| 225 | 0 | 0 | 0 |
| 0 | 225 | 0 | 0 |
| 0 | 0 | 225 | 0 |
| 0 | 0 | 0 | 225 |
| 240 | 240 | 0 | 0 |
| 191 | 190 | 0 | 0 |
| 0 | 0 | 64 | 47 |
| 0 | 0 | 0 | 177 |
| 0 | 52 | 0 | 0 |
| 51 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 54 | 1 |
| 51 | 52 | 0 | 0 |
| 191 | 190 | 0 | 0 |
| 0 | 0 | 177 | 155 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(241))| [225,0,0,0,0,225,0,0,0,0,225,0,0,0,0,225],[240,191,0,0,240,190,0,0,0,0,64,0,0,0,47,177],[0,51,0,0,52,0,0,0,0,0,240,54,0,0,0,1],[51,191,0,0,52,190,0,0,0,0,177,0,0,0,155,1] >;
C3×D20⋊7C4 in GAP, Magma, Sage, TeX
C_3\times D_{20}\rtimes_7C_4 % in TeX
G:=Group("C3xD20:7C4"); // GroupNames label
G:=SmallGroup(480,103);
// by ID
G=gap.SmallGroup(480,103);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,365,92,136,2524,1271,102,18822]);
// 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,d*b*d^-1=b^9,d*c*d^-1=b^3*c>;
// generators/relations