metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊2Dic3, C15⋊13C4≀C2, Q8⋊3(C3⋊F5), (C3×Q8)⋊2F5, (C3×D20)⋊2C4, (Q8×C15)⋊2C4, C60.36(C2×C4), (C5×Q8)⋊5Dic3, (C6×D5).34D4, C3⋊3(Q8⋊2F5), (C4×D5).30D6, C12.12(C2×F5), C12.F5⋊8C2, Q8⋊2D5.4S3, C20.4(C2×Dic3), C5⋊2(Q8⋊3Dic3), D10.7(C3⋊D4), (C3×Dic5).83D4, C6.23(C22⋊F5), C30.23(C22⋊C4), (D5×C12).71C22, Dic5.38(C3⋊D4), C10.8(C6.D4), C2.9(D10.D6), (C4×C3⋊F5)⋊7C2, C4.4(C2×C3⋊F5), (C3×Q8⋊2D5).2C2, SmallGroup(480,315)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊2Dic3
G = < a,b,c,d | a20=b2=c6=1, d2=c3, bab=a-1, cac-1=a9, dad-1=a13, cbc-1=a18b, dbd-1=a7b, dcd-1=c-1 >
Subgroups: 492 in 88 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, D4, Q8, D5, C10, Dic3, C12, C12, C2×C6, C15, C42, M4(2), C4○D4, Dic5, C20, C20, F5, D10, D10, C3⋊C8, C2×Dic3, C2×C12, C3×D4, C3×Q8, C3×D5, C30, C4≀C2, C5⋊C8, C4×D5, C4×D5, D20, D20, C5×Q8, C2×F5, C4.Dic3, C4×Dic3, C3×C4○D4, C3×Dic5, C60, C60, C3⋊F5, C6×D5, C6×D5, C4.F5, C4×F5, Q8⋊2D5, Q8⋊3Dic3, C15⋊C8, D5×C12, D5×C12, C3×D20, C3×D20, Q8×C15, C2×C3⋊F5, Q8⋊2F5, C12.F5, C4×C3⋊F5, C3×Q8⋊2D5, D20⋊2Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, F5, C2×Dic3, C3⋊D4, C4≀C2, C2×F5, C6.D4, C3⋊F5, C22⋊F5, Q8⋊3Dic3, C2×C3⋊F5, Q8⋊2F5, D10.D6, D20⋊2Dic3
►On 120 points
Generators in S120
(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 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 40)(19 39)(20 38)(41 95)(42 94)(43 93)(44 92)(45 91)(46 90)(47 89)(48 88)(49 87)(50 86)(51 85)(52 84)(53 83)(54 82)(55 81)(56 100)(57 99)(58 98)(59 97)(60 96)(61 110)(62 109)(63 108)(64 107)(65 106)(66 105)(67 104)(68 103)(69 102)(70 101)(71 120)(72 119)(73 118)(74 117)(75 116)(76 115)(77 114)(78 113)(79 112)(80 111)
(1 81 61 11 91 71)(2 90 62 20 92 80)(3 99 63 9 93 69)(4 88 64 18 94 78)(5 97 65 7 95 67)(6 86 66 16 96 76)(8 84 68 14 98 74)(10 82 70 12 100 72)(13 89 73 19 83 79)(15 87 75 17 85 77)(21 53 114 25 49 118)(22 42 115 34 50 107)(23 51 116)(24 60 117 32 52 105)(26 58 119 30 54 103)(27 47 120 39 55 112)(28 56 101)(29 45 102 37 57 110)(31 43 104 35 59 108)(33 41 106)(36 48 109 40 44 113)(38 46 111)
(1 16 11 6)(2 13 20 19)(3 10 9 12)(4 7 18 5)(8 15 14 17)(21 29 25 37)(22 26 34 30)(24 40 32 36)(27 31 39 35)(41 106)(42 103 50 119)(43 120 59 112)(44 117 48 105)(45 114 57 118)(46 111)(47 108 55 104)(49 102 53 110)(51 116)(52 113 60 109)(54 107 58 115)(56 101)(61 86 71 96)(62 83 80 89)(63 100 69 82)(64 97 78 95)(65 94 67 88)(66 91 76 81)(68 85 74 87)(70 99 72 93)(73 90 79 92)(75 84 77 98)
G:=sub<Sym(120)| (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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,40)(19,39)(20,38)(41,95)(42,94)(43,93)(44,92)(45,91)(46,90)(47,89)(48,88)(49,87)(50,86)(51,85)(52,84)(53,83)(54,82)(55,81)(56,100)(57,99)(58,98)(59,97)(60,96)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,120)(72,119)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,112)(80,111), (1,81,61,11,91,71)(2,90,62,20,92,80)(3,99,63,9,93,69)(4,88,64,18,94,78)(5,97,65,7,95,67)(6,86,66,16,96,76)(8,84,68,14,98,74)(10,82,70,12,100,72)(13,89,73,19,83,79)(15,87,75,17,85,77)(21,53,114,25,49,118)(22,42,115,34,50,107)(23,51,116)(24,60,117,32,52,105)(26,58,119,30,54,103)(27,47,120,39,55,112)(28,56,101)(29,45,102,37,57,110)(31,43,104,35,59,108)(33,41,106)(36,48,109,40,44,113)(38,46,111), (1,16,11,6)(2,13,20,19)(3,10,9,12)(4,7,18,5)(8,15,14,17)(21,29,25,37)(22,26,34,30)(24,40,32,36)(27,31,39,35)(41,106)(42,103,50,119)(43,120,59,112)(44,117,48,105)(45,114,57,118)(46,111)(47,108,55,104)(49,102,53,110)(51,116)(52,113,60,109)(54,107,58,115)(56,101)(61,86,71,96)(62,83,80,89)(63,100,69,82)(64,97,78,95)(65,94,67,88)(66,91,76,81)(68,85,74,87)(70,99,72,93)(73,90,79,92)(75,84,77,98)>;
G:=Group( (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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,40)(19,39)(20,38)(41,95)(42,94)(43,93)(44,92)(45,91)(46,90)(47,89)(48,88)(49,87)(50,86)(51,85)(52,84)(53,83)(54,82)(55,81)(56,100)(57,99)(58,98)(59,97)(60,96)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,120)(72,119)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,112)(80,111), (1,81,61,11,91,71)(2,90,62,20,92,80)(3,99,63,9,93,69)(4,88,64,18,94,78)(5,97,65,7,95,67)(6,86,66,16,96,76)(8,84,68,14,98,74)(10,82,70,12,100,72)(13,89,73,19,83,79)(15,87,75,17,85,77)(21,53,114,25,49,118)(22,42,115,34,50,107)(23,51,116)(24,60,117,32,52,105)(26,58,119,30,54,103)(27,47,120,39,55,112)(28,56,101)(29,45,102,37,57,110)(31,43,104,35,59,108)(33,41,106)(36,48,109,40,44,113)(38,46,111), (1,16,11,6)(2,13,20,19)(3,10,9,12)(4,7,18,5)(8,15,14,17)(21,29,25,37)(22,26,34,30)(24,40,32,36)(27,31,39,35)(41,106)(42,103,50,119)(43,120,59,112)(44,117,48,105)(45,114,57,118)(46,111)(47,108,55,104)(49,102,53,110)(51,116)(52,113,60,109)(54,107,58,115)(56,101)(61,86,71,96)(62,83,80,89)(63,100,69,82)(64,97,78,95)(65,94,67,88)(66,91,76,81)(68,85,74,87)(70,99,72,93)(73,90,79,92)(75,84,77,98) );
G=PermutationGroup([[(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)], [(1,37),(2,36),(3,35),(4,34),(5,33),(6,32),(7,31),(8,30),(9,29),(10,28),(11,27),(12,26),(13,25),(14,24),(15,23),(16,22),(17,21),(18,40),(19,39),(20,38),(41,95),(42,94),(43,93),(44,92),(45,91),(46,90),(47,89),(48,88),(49,87),(50,86),(51,85),(52,84),(53,83),(54,82),(55,81),(56,100),(57,99),(58,98),(59,97),(60,96),(61,110),(62,109),(63,108),(64,107),(65,106),(66,105),(67,104),(68,103),(69,102),(70,101),(71,120),(72,119),(73,118),(74,117),(75,116),(76,115),(77,114),(78,113),(79,112),(80,111)], [(1,81,61,11,91,71),(2,90,62,20,92,80),(3,99,63,9,93,69),(4,88,64,18,94,78),(5,97,65,7,95,67),(6,86,66,16,96,76),(8,84,68,14,98,74),(10,82,70,12,100,72),(13,89,73,19,83,79),(15,87,75,17,85,77),(21,53,114,25,49,118),(22,42,115,34,50,107),(23,51,116),(24,60,117,32,52,105),(26,58,119,30,54,103),(27,47,120,39,55,112),(28,56,101),(29,45,102,37,57,110),(31,43,104,35,59,108),(33,41,106),(36,48,109,40,44,113),(38,46,111)], [(1,16,11,6),(2,13,20,19),(3,10,9,12),(4,7,18,5),(8,15,14,17),(21,29,25,37),(22,26,34,30),(24,40,32,36),(27,31,39,35),(41,106),(42,103,50,119),(43,120,59,112),(44,117,48,105),(45,114,57,118),(46,111),(47,108,55,104),(49,102,53,110),(51,116),(52,113,60,109),(54,107,58,115),(56,101),(61,86,71,96),(62,83,80,89),(63,100,69,82),(64,97,78,95),(65,94,67,88),(66,91,76,81),(68,85,74,87),(70,99,72,93),(73,90,79,92),(75,84,77,98)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 6A | 6B | 6C | 6D | 8A | 8B | 10 | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | 20B | 20C | 30A | 30B | 60A | ··· | 60F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 10 | 20 | 2 | 2 | 4 | 5 | 5 | 30 | 30 | 30 | 30 | 4 | 2 | 20 | 20 | 20 | 60 | 60 | 4 | 4 | 4 | 4 | 10 | 10 | 4 | 4 | 8 | 8 | 8 | 4 | 4 | 8 | ··· | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | - | - | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | D6 | Dic3 | Dic3 | C3⋊D4 | C3⋊D4 | C4≀C2 | F5 | C2×F5 | C3⋊F5 | C22⋊F5 | Q8⋊3Dic3 | C2×C3⋊F5 | D10.D6 | Q8⋊2F5 | D20⋊2Dic3 |
| kernel | D20⋊2Dic3 | C12.F5 | C4×C3⋊F5 | C3×Q8⋊2D5 | C3×D20 | Q8×C15 | Q8⋊2D5 | C3×Dic5 | C6×D5 | C4×D5 | D20 | C5×Q8 | Dic5 | D10 | C15 | C3×Q8 | C12 | Q8 | C6 | C5 | C4 | C2 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 2 |
Matrix representation of D20⋊2Dic3 ►in GL8(𝔽241)
| 64 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 | 0 | 0 |
| 103 | 0 | 177 | 0 | 0 | 0 | 0 | 0 |
| 39 | 64 | 0 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 0 |
| 85 | 0 | 2 | 0 | 0 | 0 | 0 | 0 |
| 86 | 240 | 0 | 2 | 0 | 0 | 0 | 0 |
| 3 | 0 | 156 | 0 | 0 | 0 | 0 | 0 |
| 3 | 0 | 155 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 1 | 240 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 43 | 42 | 0 | 1 | 0 | 0 | 0 | 0 |
| 156 | 44 | 240 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 229 | 0 | 12 | 114 |
| 0 | 0 | 0 | 0 | 229 | 12 | 126 | 0 |
| 0 | 0 | 0 | 0 | 0 | 126 | 12 | 229 |
| 0 | 0 | 0 | 0 | 114 | 12 | 0 | 229 |
| 177 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 64 | 64 | 0 | 0 | 0 | 0 | 0 | 0 |
| 232 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 168 | 153 | 240 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 229 | 12 | 127 | 0 |
| 0 | 0 | 0 | 0 | 115 | 12 | 0 | 229 |
| 0 | 0 | 0 | 0 | 229 | 0 | 12 | 115 |
| 0 | 0 | 0 | 0 | 0 | 127 | 12 | 229 |
G:=sub<GL(8,GF(241))| [64,0,103,39,0,0,0,0,0,64,0,64,0,0,0,0,0,0,177,0,0,0,0,0,0,0,0,177,0,0,0,0,0,0,0,0,240,240,240,240,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[85,86,3,3,0,0,0,0,0,240,0,0,0,0,0,0,2,0,156,155,0,0,0,0,0,2,0,1,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0],[0,1,43,156,0,0,0,0,240,1,42,44,0,0,0,0,0,0,0,240,0,0,0,0,0,0,1,240,0,0,0,0,0,0,0,0,229,229,0,114,0,0,0,0,0,12,126,12,0,0,0,0,12,126,12,0,0,0,0,0,114,0,229,229],[177,64,232,168,0,0,0,0,0,64,0,153,0,0,0,0,0,0,1,240,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,229,115,229,0,0,0,0,0,12,12,0,127,0,0,0,0,127,0,12,12,0,0,0,0,0,229,115,229] >;
D20⋊2Dic3 in GAP, Magma, Sage, TeX
D_{20}\rtimes_2{\rm Dic}_3 % in TeX
G:=Group("D20:2Dic3"); // GroupNames label
G:=SmallGroup(480,315);
// by ID
G=gap.SmallGroup(480,315);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,120,100,675,346,80,2693,14118,4724]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^6=1,d^2=c^3,b*a*b=a^-1,c*a*c^-1=a^9,d*a*d^-1=a^13,c*b*c^-1=a^18*b,d*b*d^-1=a^7*b,d*c*d^-1=c^-1>;
// generators/relations