direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6×D20, C30⋊5D4, C12⋊9D10, C60⋊10C22, C30.40C23, C5⋊1(C6×D4), C4⋊2(C6×D5), (C2×C20)⋊3C6, C20⋊2(C2×C6), (C2×C60)⋊8C2, (C2×C12)⋊6D5, C10⋊1(C3×D4), C15⋊11(C2×D4), D10⋊1(C2×C6), (C2×C6).38D10, (C6×D5)⋊9C22, (C22×D5)⋊2C6, C10.3(C22×C6), C6.40(C22×D5), C22.10(C6×D5), (C2×C30).39C22, (D5×C2×C6)⋊5C2, C2.4(D5×C2×C6), (C2×C4)⋊2(C3×D5), (C2×C10).10(C2×C6), SmallGroup(240,157)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×D20
G = < a,b,c | a6=b20=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 356 in 108 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, C6, C6, C6, C2×C4, D4, C23, D5, C10, C10, C12, C2×C6, C2×C6, C15, C2×D4, C20, D10, D10, C2×C10, C2×C12, C3×D4, C22×C6, C3×D5, C30, C30, D20, C2×C20, C22×D5, C6×D4, C60, C6×D5, C6×D5, C2×C30, C2×D20, C3×D20, C2×C60, D5×C2×C6, C6×D20
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, D10, C3×D4, C22×C6, C3×D5, D20, C22×D5, C6×D4, C6×D5, C2×D20, C3×D20, D5×C2×C6, C6×D20
►On 120 points
Generators in S120
(1 104 70 83 52 31)(2 105 71 84 53 32)(3 106 72 85 54 33)(4 107 73 86 55 34)(5 108 74 87 56 35)(6 109 75 88 57 36)(7 110 76 89 58 37)(8 111 77 90 59 38)(9 112 78 91 60 39)(10 113 79 92 41 40)(11 114 80 93 42 21)(12 115 61 94 43 22)(13 116 62 95 44 23)(14 117 63 96 45 24)(15 118 64 97 46 25)(16 119 65 98 47 26)(17 120 66 99 48 27)(18 101 67 100 49 28)(19 102 68 81 50 29)(20 103 69 82 51 30)
(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 92)(2 91)(3 90)(4 89)(5 88)(6 87)(7 86)(8 85)(9 84)(10 83)(11 82)(12 81)(13 100)(14 99)(15 98)(16 97)(17 96)(18 95)(19 94)(20 93)(21 69)(22 68)(23 67)(24 66)(25 65)(26 64)(27 63)(28 62)(29 61)(30 80)(31 79)(32 78)(33 77)(34 76)(35 75)(36 74)(37 73)(38 72)(39 71)(40 70)(41 104)(42 103)(43 102)(44 101)(45 120)(46 119)(47 118)(48 117)(49 116)(50 115)(51 114)(52 113)(53 112)(54 111)(55 110)(56 109)(57 108)(58 107)(59 106)(60 105)
G:=sub<Sym(120)| (1,104,70,83,52,31)(2,105,71,84,53,32)(3,106,72,85,54,33)(4,107,73,86,55,34)(5,108,74,87,56,35)(6,109,75,88,57,36)(7,110,76,89,58,37)(8,111,77,90,59,38)(9,112,78,91,60,39)(10,113,79,92,41,40)(11,114,80,93,42,21)(12,115,61,94,43,22)(13,116,62,95,44,23)(14,117,63,96,45,24)(15,118,64,97,46,25)(16,119,65,98,47,26)(17,120,66,99,48,27)(18,101,67,100,49,28)(19,102,68,81,50,29)(20,103,69,82,51,30), (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), (1,92)(2,91)(3,90)(4,89)(5,88)(6,87)(7,86)(8,85)(9,84)(10,83)(11,82)(12,81)(13,100)(14,99)(15,98)(16,97)(17,96)(18,95)(19,94)(20,93)(21,69)(22,68)(23,67)(24,66)(25,65)(26,64)(27,63)(28,62)(29,61)(30,80)(31,79)(32,78)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)(40,70)(41,104)(42,103)(43,102)(44,101)(45,120)(46,119)(47,118)(48,117)(49,116)(50,115)(51,114)(52,113)(53,112)(54,111)(55,110)(56,109)(57,108)(58,107)(59,106)(60,105)>;
G:=Group( (1,104,70,83,52,31)(2,105,71,84,53,32)(3,106,72,85,54,33)(4,107,73,86,55,34)(5,108,74,87,56,35)(6,109,75,88,57,36)(7,110,76,89,58,37)(8,111,77,90,59,38)(9,112,78,91,60,39)(10,113,79,92,41,40)(11,114,80,93,42,21)(12,115,61,94,43,22)(13,116,62,95,44,23)(14,117,63,96,45,24)(15,118,64,97,46,25)(16,119,65,98,47,26)(17,120,66,99,48,27)(18,101,67,100,49,28)(19,102,68,81,50,29)(20,103,69,82,51,30), (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), (1,92)(2,91)(3,90)(4,89)(5,88)(6,87)(7,86)(8,85)(9,84)(10,83)(11,82)(12,81)(13,100)(14,99)(15,98)(16,97)(17,96)(18,95)(19,94)(20,93)(21,69)(22,68)(23,67)(24,66)(25,65)(26,64)(27,63)(28,62)(29,61)(30,80)(31,79)(32,78)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)(40,70)(41,104)(42,103)(43,102)(44,101)(45,120)(46,119)(47,118)(48,117)(49,116)(50,115)(51,114)(52,113)(53,112)(54,111)(55,110)(56,109)(57,108)(58,107)(59,106)(60,105) );
G=PermutationGroup([[(1,104,70,83,52,31),(2,105,71,84,53,32),(3,106,72,85,54,33),(4,107,73,86,55,34),(5,108,74,87,56,35),(6,109,75,88,57,36),(7,110,76,89,58,37),(8,111,77,90,59,38),(9,112,78,91,60,39),(10,113,79,92,41,40),(11,114,80,93,42,21),(12,115,61,94,43,22),(13,116,62,95,44,23),(14,117,63,96,45,24),(15,118,64,97,46,25),(16,119,65,98,47,26),(17,120,66,99,48,27),(18,101,67,100,49,28),(19,102,68,81,50,29),(20,103,69,82,51,30)], [(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)], [(1,92),(2,91),(3,90),(4,89),(5,88),(6,87),(7,86),(8,85),(9,84),(10,83),(11,82),(12,81),(13,100),(14,99),(15,98),(16,97),(17,96),(18,95),(19,94),(20,93),(21,69),(22,68),(23,67),(24,66),(25,65),(26,64),(27,63),(28,62),(29,61),(30,80),(31,79),(32,78),(33,77),(34,76),(35,75),(36,74),(37,73),(38,72),(39,71),(40,70),(41,104),(42,103),(43,102),(44,101),(45,120),(46,119),(47,118),(48,117),(49,116),(50,115),(51,114),(52,113),(53,112),(54,111),(55,110),(56,109),(57,108),(58,107),(59,106),(60,105)]])
C6×D20 is a maximal subgroup of
C60.28D4 C30.D8 C6.D40 (C2×C60)⋊C4 C60.36D4 D60⋊30C22 C60.44D4 C60.88D4 (C6×D5).D4 Dic15⋊D4 Dic3⋊D20 D20⋊8Dic3 C60⋊4D4 C12⋊D20 C60⋊10D4 C12⋊2D20 D6⋊4D20 D20⋊25D6 C6×D4×D5
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 5A | 5B | 6A | ··· | 6F | 6G | ··· | 6N | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | D5 | D10 | D10 | C3×D4 | C3×D5 | D20 | C6×D5 | C6×D5 | C3×D20 |
| kernel | C6×D20 | C3×D20 | C2×C60 | D5×C2×C6 | C2×D20 | D20 | C2×C20 | C22×D5 | C30 | C2×C12 | C12 | C2×C6 | C10 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 2 | 8 | 2 | 4 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 4 | 16 |
Matrix representation of C6×D20 ►in GL3(𝔽61) generated by
| 14 | 0 | 0 |
| 0 | 14 | 0 |
| 0 | 0 | 14 |
| 60 | 0 | 0 |
| 0 | 59 | 29 |
| 0 | 32 | 54 |
| 60 | 0 | 0 |
| 0 | 54 | 32 |
| 0 | 29 | 7 |
G:=sub<GL(3,GF(61))| [14,0,0,0,14,0,0,0,14],[60,0,0,0,59,32,0,29,54],[60,0,0,0,54,29,0,32,7] >;
C6×D20 in GAP, Magma, Sage, TeX
C_6\times D_{20} % in TeX
G:=Group("C6xD20"); // GroupNames label
G:=SmallGroup(240,157);
// by ID
G=gap.SmallGroup(240,157);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-5,506,122,6917]);
// Polycyclic
G:=Group<a,b,c|a^6=b^20=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations