metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C6).2D20, C15⋊7(C23⋊C4), (C6×Dic5)⋊1C4, (C2×C30).36D4, C23.8(S3×D5), C6.D4⋊1D5, (C2×Dic5)⋊1Dic3, (C22×D5)⋊3Dic3, (C22×C10).27D6, (C22×C6).12D10, C5⋊3(C23.7D6), C30.38D4⋊14C2, C22.5(D5×Dic3), C30.66(C22⋊C4), C22.8(C15⋊D4), C22.8(C3⋊D20), C3⋊3(C23.1D10), C6.32(D10⋊C4), (C22×C30).26C22, C10.22(C6.D4), C2.11(D10⋊Dic3), (D5×C2×C6)⋊1C4, (C2×C6).49(C4×D5), (C6×C5⋊D4).1C2, (C2×C5⋊D4).1S3, (C2×C30).90(C2×C4), (C2×C6).4(C5⋊D4), (C5×C6.D4)⋊1C2, (C2×C10).50(C3⋊D4), (C2×C10).23(C2×Dic3), SmallGroup(480,71)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C6).D20
G = < a,b,c,d | a2=b6=c20=1, d2=a, ab=ba, cac-1=ab3, ad=da, cbc-1=dbd-1=b-1, dcd-1=ac-1 >
Subgroups: 540 in 104 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2 [×4], C3, C4 [×3], C22 [×3], C22 [×3], C5, C6, C6 [×4], C2×C4 [×3], D4 [×2], C23, C23, D5, C10, C10 [×3], Dic3 [×2], C12, C2×C6 [×3], C2×C6 [×3], C15, C22⋊C4 [×2], C2×D4, Dic5 [×2], C20, D10 [×2], C2×C10 [×3], C2×C10, C2×Dic3 [×2], C2×C12, C3×D4 [×2], C22×C6, C22×C6, C3×D5, C30, C30 [×3], C23⋊C4, C2×Dic5, C2×Dic5, C5⋊D4 [×2], C2×C20, C22×D5, C22×C10, C6.D4, C6.D4, C6×D4, C5×Dic3, C3×Dic5, Dic15, C6×D5 [×2], C2×C30 [×3], C2×C30, C23.D5, C5×C22⋊C4, C2×C5⋊D4, C23.7D6, C6×Dic5, C3×C5⋊D4 [×2], C10×Dic3, C2×Dic15, D5×C2×C6, C22×C30, C23.1D10, C5×C6.D4, C30.38D4, C6×C5⋊D4, (C2×C6).D20
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, Dic3 [×2], D6, C22⋊C4, D10, C2×Dic3, C3⋊D4 [×2], C23⋊C4, C4×D5, D20, C5⋊D4, C6.D4, S3×D5, D10⋊C4, C23.7D6, D5×Dic3, C15⋊D4, C3⋊D20, C23.1D10, D10⋊Dic3, (C2×C6).D20
►On 120 points
(1 27)(3 29)(5 31)(7 33)(9 35)(11 37)(13 39)(15 21)(17 23)(19 25)(42 75)(44 77)(46 79)(48 61)(50 63)(52 65)(54 67)(56 69)(58 71)(60 73)(81 102)(83 104)(85 106)(87 108)(89 110)(91 112)(93 114)(95 116)(97 118)(99 120)
(1 58 97 27 71 118)(2 119 72 28 98 59)(3 60 99 29 73 120)(4 101 74 30 100 41)(5 42 81 31 75 102)(6 103 76 32 82 43)(7 44 83 33 77 104)(8 105 78 34 84 45)(9 46 85 35 79 106)(10 107 80 36 86 47)(11 48 87 37 61 108)(12 109 62 38 88 49)(13 50 89 39 63 110)(14 111 64 40 90 51)(15 52 91 21 65 112)(16 113 66 22 92 53)(17 54 93 23 67 114)(18 115 68 24 94 55)(19 56 95 25 69 116)(20 117 70 26 96 57)
(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 21 27 15)(2 40)(3 13 29 39)(4 12)(5 37 31 11)(6 36)(7 9 33 35)(10 32)(14 28)(16 20)(17 25 23 19)(18 24)(22 26)(30 38)(41 109)(42 87 75 108)(43 86)(44 106 77 85)(45 105)(46 83 79 104)(47 82)(48 102 61 81)(49 101)(50 99 63 120)(51 98)(52 118 65 97)(53 117)(54 95 67 116)(55 94)(56 114 69 93)(57 113)(58 91 71 112)(59 90)(60 110 73 89)(62 100)(64 119)(66 96)(68 115)(70 92)(72 111)(74 88)(76 107)(78 84)(80 103)
G:=sub<Sym(120)| (1,27)(3,29)(5,31)(7,33)(9,35)(11,37)(13,39)(15,21)(17,23)(19,25)(42,75)(44,77)(46,79)(48,61)(50,63)(52,65)(54,67)(56,69)(58,71)(60,73)(81,102)(83,104)(85,106)(87,108)(89,110)(91,112)(93,114)(95,116)(97,118)(99,120), (1,58,97,27,71,118)(2,119,72,28,98,59)(3,60,99,29,73,120)(4,101,74,30,100,41)(5,42,81,31,75,102)(6,103,76,32,82,43)(7,44,83,33,77,104)(8,105,78,34,84,45)(9,46,85,35,79,106)(10,107,80,36,86,47)(11,48,87,37,61,108)(12,109,62,38,88,49)(13,50,89,39,63,110)(14,111,64,40,90,51)(15,52,91,21,65,112)(16,113,66,22,92,53)(17,54,93,23,67,114)(18,115,68,24,94,55)(19,56,95,25,69,116)(20,117,70,26,96,57), (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), (1,21,27,15)(2,40)(3,13,29,39)(4,12)(5,37,31,11)(6,36)(7,9,33,35)(10,32)(14,28)(16,20)(17,25,23,19)(18,24)(22,26)(30,38)(41,109)(42,87,75,108)(43,86)(44,106,77,85)(45,105)(46,83,79,104)(47,82)(48,102,61,81)(49,101)(50,99,63,120)(51,98)(52,118,65,97)(53,117)(54,95,67,116)(55,94)(56,114,69,93)(57,113)(58,91,71,112)(59,90)(60,110,73,89)(62,100)(64,119)(66,96)(68,115)(70,92)(72,111)(74,88)(76,107)(78,84)(80,103)>;
G:=Group( (1,27)(3,29)(5,31)(7,33)(9,35)(11,37)(13,39)(15,21)(17,23)(19,25)(42,75)(44,77)(46,79)(48,61)(50,63)(52,65)(54,67)(56,69)(58,71)(60,73)(81,102)(83,104)(85,106)(87,108)(89,110)(91,112)(93,114)(95,116)(97,118)(99,120), (1,58,97,27,71,118)(2,119,72,28,98,59)(3,60,99,29,73,120)(4,101,74,30,100,41)(5,42,81,31,75,102)(6,103,76,32,82,43)(7,44,83,33,77,104)(8,105,78,34,84,45)(9,46,85,35,79,106)(10,107,80,36,86,47)(11,48,87,37,61,108)(12,109,62,38,88,49)(13,50,89,39,63,110)(14,111,64,40,90,51)(15,52,91,21,65,112)(16,113,66,22,92,53)(17,54,93,23,67,114)(18,115,68,24,94,55)(19,56,95,25,69,116)(20,117,70,26,96,57), (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), (1,21,27,15)(2,40)(3,13,29,39)(4,12)(5,37,31,11)(6,36)(7,9,33,35)(10,32)(14,28)(16,20)(17,25,23,19)(18,24)(22,26)(30,38)(41,109)(42,87,75,108)(43,86)(44,106,77,85)(45,105)(46,83,79,104)(47,82)(48,102,61,81)(49,101)(50,99,63,120)(51,98)(52,118,65,97)(53,117)(54,95,67,116)(55,94)(56,114,69,93)(57,113)(58,91,71,112)(59,90)(60,110,73,89)(62,100)(64,119)(66,96)(68,115)(70,92)(72,111)(74,88)(76,107)(78,84)(80,103) );
G=PermutationGroup([(1,27),(3,29),(5,31),(7,33),(9,35),(11,37),(13,39),(15,21),(17,23),(19,25),(42,75),(44,77),(46,79),(48,61),(50,63),(52,65),(54,67),(56,69),(58,71),(60,73),(81,102),(83,104),(85,106),(87,108),(89,110),(91,112),(93,114),(95,116),(97,118),(99,120)], [(1,58,97,27,71,118),(2,119,72,28,98,59),(3,60,99,29,73,120),(4,101,74,30,100,41),(5,42,81,31,75,102),(6,103,76,32,82,43),(7,44,83,33,77,104),(8,105,78,34,84,45),(9,46,85,35,79,106),(10,107,80,36,86,47),(11,48,87,37,61,108),(12,109,62,38,88,49),(13,50,89,39,63,110),(14,111,64,40,90,51),(15,52,91,21,65,112),(16,113,66,22,92,53),(17,54,93,23,67,114),(18,115,68,24,94,55),(19,56,95,25,69,116),(20,117,70,26,96,57)], [(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)], [(1,21,27,15),(2,40),(3,13,29,39),(4,12),(5,37,31,11),(6,36),(7,9,33,35),(10,32),(14,28),(16,20),(17,25,23,19),(18,24),(22,26),(30,38),(41,109),(42,87,75,108),(43,86),(44,106,77,85),(45,105),(46,83,79,104),(47,82),(48,102,61,81),(49,101),(50,99,63,120),(51,98),(52,118,65,97),(53,117),(54,95,67,116),(55,94),(56,114,69,93),(57,113),(58,91,71,112),(59,90),(60,110,73,89),(62,100),(64,119),(66,96),(68,115),(70,92),(72,111),(74,88),(76,107),(78,84),(80,103)])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 2 | 2 | 2 | 20 | 2 | 12 | 12 | 20 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | + | + | + | + | + | - | - | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D5 | Dic3 | Dic3 | D6 | D10 | C3⋊D4 | C4×D5 | D20 | C5⋊D4 | C23⋊C4 | S3×D5 | C23.7D6 | D5×Dic3 | C15⋊D4 | C3⋊D20 | C23.1D10 | (C2×C6).D20 |
| kernel | (C2×C6).D20 | C5×C6.D4 | C30.38D4 | C6×C5⋊D4 | C6×Dic5 | D5×C2×C6 | C2×C5⋊D4 | C2×C30 | C6.D4 | C2×Dic5 | C22×D5 | C22×C10 | C22×C6 | C2×C10 | C2×C6 | C2×C6 | C2×C6 | C15 | C23 | C5 | C22 | C22 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of (C2×C6).D20 ►in GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 33 | 44 | 1 | 0 |
| 33 | 44 | 0 | 1 |
| 25 | 45 | 0 | 0 |
| 49 | 37 | 0 | 0 |
| 14 | 53 | 53 | 16 |
| 41 | 28 | 45 | 9 |
| 36 | 48 | 60 | 42 |
| 59 | 38 | 18 | 17 |
| 60 | 58 | 17 | 31 |
| 13 | 27 | 17 | 31 |
| 47 | 44 | 0 | 0 |
| 8 | 14 | 0 | 0 |
| 41 | 25 | 0 | 1 |
| 41 | 25 | 1 | 0 |
G:=sub<GL(4,GF(61))| [60,0,33,33,0,60,44,44,0,0,1,0,0,0,0,1],[25,49,14,41,45,37,53,28,0,0,53,45,0,0,16,9],[36,59,60,13,48,38,58,27,60,18,17,17,42,17,31,31],[47,8,41,41,44,14,25,25,0,0,0,1,0,0,1,0] >;
(C2×C6).D20 in GAP, Magma, Sage, TeX
(C_2\times C_6).D_{20} % in TeX
G:=Group("(C2xC6).D20"); // GroupNames label
G:=SmallGroup(480,71);
// by ID
G=gap.SmallGroup(480,71);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,36,422,346,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^20=1,d^2=a,a*b=b*a,c*a*c^-1=a*b^3,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a*c^-1>;
// generators/relations