metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30.8D4, D20.6D6, D12.6D10, C60.6C23, Dic15.40D4, Dic30⋊1C22, C3⋊C8⋊6D10, D4⋊D5⋊3S3, D4⋊S3⋊3D5, D4⋊6(S3×D5), (C5×D4)⋊4D6, C5⋊2C8⋊6D6, (C3×D4)⋊4D10, C5⋊3(D8⋊S3), C6.68(D4×D5), C20⋊D6⋊2C2, C3⋊3(D8⋊D5), C10.69(S3×D4), D4⋊2D15⋊1C2, C15⋊15(C8⋊C22), D12.D5⋊1C2, C6.D20⋊1C2, C30.168(C2×D4), (D4×C15)⋊6C22, C20.6(C22×S3), C12.6(C22×D5), D30.5C4⋊1C2, (C5×D12).2C22, (C4×D15).2C22, (C3×D20).2C22, C2.21(D10⋊D6), C4.6(C2×S3×D5), (C3×D4⋊D5)⋊4C2, (C5×D4⋊S3)⋊4C2, (C5×C3⋊C8)⋊4C22, (C3×C5⋊2C8)⋊4C22, SmallGroup(480,558)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30.8D4
G = < a,b,c,d | a20=b2=c6=1, d2=a5, bab=a-1, cac-1=a11, ad=da, cbc-1=dbd-1=a15b, dcd-1=a5c-1 >
Subgroups: 940 in 136 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, D4, Q8, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, C20, D10, C2×C10, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C3×D4, C22×S3, C5×S3, C3×D5, D15, C30, C30, C8⋊C22, C5⋊2C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C5×D4, C5×D4, C22×D5, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×D8, S3×D4, D4⋊2S3, Dic15, Dic15, C60, S3×D5, C6×D5, S3×C10, D30, C2×C30, C8⋊D5, C40⋊C2, D4⋊D5, D4.D5, C5×D8, D4×D5, D4⋊2D5, D8⋊S3, C5×C3⋊C8, C3×C5⋊2C8, C15⋊D4, C3×D20, C5×D12, Dic30, C4×D15, C2×Dic15, C15⋊7D4, D4×C15, C2×S3×D5, D8⋊D5, D30.5C4, C6.D20, D12.D5, C3×D4⋊D5, C5×D4⋊S3, C20⋊D6, D4⋊2D15, D30.8D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C8⋊C22, C22×D5, S3×D4, S3×D5, D4×D5, D8⋊S3, C2×S3×D5, D8⋊D5, D10⋊D6, D30.8D4
►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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(33 40)(34 39)(35 38)(36 37)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 48)(55 60)(56 59)(57 58)(61 63)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(81 87)(82 86)(83 85)(88 100)(89 99)(90 98)(91 97)(92 96)(93 95)(101 113)(102 112)(103 111)(104 110)(105 109)(106 108)(114 120)(115 119)(116 118)
(1 105 48 80 27 92)(2 116 49 71 28 83)(3 107 50 62 29 94)(4 118 51 73 30 85)(5 109 52 64 31 96)(6 120 53 75 32 87)(7 111 54 66 33 98)(8 102 55 77 34 89)(9 113 56 68 35 100)(10 104 57 79 36 91)(11 115 58 70 37 82)(12 106 59 61 38 93)(13 117 60 72 39 84)(14 108 41 63 40 95)(15 119 42 74 21 86)(16 110 43 65 22 97)(17 101 44 76 23 88)(18 112 45 67 24 99)(19 103 46 78 25 90)(20 114 47 69 26 81)
(1 92 6 97 11 82 16 87)(2 93 7 98 12 83 17 88)(3 94 8 99 13 84 18 89)(4 95 9 100 14 85 19 90)(5 96 10 81 15 86 20 91)(21 119 26 104 31 109 36 114)(22 120 27 105 32 110 37 115)(23 101 28 106 33 111 38 116)(24 102 29 107 34 112 39 117)(25 103 30 108 35 113 40 118)(41 73 46 78 51 63 56 68)(42 74 47 79 52 64 57 69)(43 75 48 80 53 65 58 70)(44 76 49 61 54 66 59 71)(45 77 50 62 55 67 60 72)
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(33,40)(34,39)(35,38)(36,37)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(55,60)(56,59)(57,58)(61,63)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(81,87)(82,86)(83,85)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95)(101,113)(102,112)(103,111)(104,110)(105,109)(106,108)(114,120)(115,119)(116,118), (1,105,48,80,27,92)(2,116,49,71,28,83)(3,107,50,62,29,94)(4,118,51,73,30,85)(5,109,52,64,31,96)(6,120,53,75,32,87)(7,111,54,66,33,98)(8,102,55,77,34,89)(9,113,56,68,35,100)(10,104,57,79,36,91)(11,115,58,70,37,82)(12,106,59,61,38,93)(13,117,60,72,39,84)(14,108,41,63,40,95)(15,119,42,74,21,86)(16,110,43,65,22,97)(17,101,44,76,23,88)(18,112,45,67,24,99)(19,103,46,78,25,90)(20,114,47,69,26,81), (1,92,6,97,11,82,16,87)(2,93,7,98,12,83,17,88)(3,94,8,99,13,84,18,89)(4,95,9,100,14,85,19,90)(5,96,10,81,15,86,20,91)(21,119,26,104,31,109,36,114)(22,120,27,105,32,110,37,115)(23,101,28,106,33,111,38,116)(24,102,29,107,34,112,39,117)(25,103,30,108,35,113,40,118)(41,73,46,78,51,63,56,68)(42,74,47,79,52,64,57,69)(43,75,48,80,53,65,58,70)(44,76,49,61,54,66,59,71)(45,77,50,62,55,67,60,72)>;
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(33,40)(34,39)(35,38)(36,37)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(55,60)(56,59)(57,58)(61,63)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(81,87)(82,86)(83,85)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95)(101,113)(102,112)(103,111)(104,110)(105,109)(106,108)(114,120)(115,119)(116,118), (1,105,48,80,27,92)(2,116,49,71,28,83)(3,107,50,62,29,94)(4,118,51,73,30,85)(5,109,52,64,31,96)(6,120,53,75,32,87)(7,111,54,66,33,98)(8,102,55,77,34,89)(9,113,56,68,35,100)(10,104,57,79,36,91)(11,115,58,70,37,82)(12,106,59,61,38,93)(13,117,60,72,39,84)(14,108,41,63,40,95)(15,119,42,74,21,86)(16,110,43,65,22,97)(17,101,44,76,23,88)(18,112,45,67,24,99)(19,103,46,78,25,90)(20,114,47,69,26,81), (1,92,6,97,11,82,16,87)(2,93,7,98,12,83,17,88)(3,94,8,99,13,84,18,89)(4,95,9,100,14,85,19,90)(5,96,10,81,15,86,20,91)(21,119,26,104,31,109,36,114)(22,120,27,105,32,110,37,115)(23,101,28,106,33,111,38,116)(24,102,29,107,34,112,39,117)(25,103,30,108,35,113,40,118)(41,73,46,78,51,63,56,68)(42,74,47,79,52,64,57,69)(43,75,48,80,53,65,58,70)(44,76,49,61,54,66,59,71)(45,77,50,62,55,67,60,72) );
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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,32),(22,31),(23,30),(24,29),(25,28),(26,27),(33,40),(34,39),(35,38),(36,37),(41,54),(42,53),(43,52),(44,51),(45,50),(46,49),(47,48),(55,60),(56,59),(57,58),(61,63),(64,80),(65,79),(66,78),(67,77),(68,76),(69,75),(70,74),(71,73),(81,87),(82,86),(83,85),(88,100),(89,99),(90,98),(91,97),(92,96),(93,95),(101,113),(102,112),(103,111),(104,110),(105,109),(106,108),(114,120),(115,119),(116,118)], [(1,105,48,80,27,92),(2,116,49,71,28,83),(3,107,50,62,29,94),(4,118,51,73,30,85),(5,109,52,64,31,96),(6,120,53,75,32,87),(7,111,54,66,33,98),(8,102,55,77,34,89),(9,113,56,68,35,100),(10,104,57,79,36,91),(11,115,58,70,37,82),(12,106,59,61,38,93),(13,117,60,72,39,84),(14,108,41,63,40,95),(15,119,42,74,21,86),(16,110,43,65,22,97),(17,101,44,76,23,88),(18,112,45,67,24,99),(19,103,46,78,25,90),(20,114,47,69,26,81)], [(1,92,6,97,11,82,16,87),(2,93,7,98,12,83,17,88),(3,94,8,99,13,84,18,89),(4,95,9,100,14,85,19,90),(5,96,10,81,15,86,20,91),(21,119,26,104,31,109,36,114),(22,120,27,105,32,110,37,115),(23,101,28,106,33,111,38,116),(24,102,29,107,34,112,39,117),(25,103,30,108,35,113,40,118),(41,73,46,78,51,63,56,68),(42,74,47,79,52,64,57,69),(43,75,48,80,53,65,58,70),(44,76,49,61,54,66,59,71),(45,77,50,62,55,67,60,72)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 12 | 15A | 15B | 20A | 20B | 24A | 24B | 30A | 30B | 30C | 30D | 30E | 30F | 40A | 40B | 40C | 40D | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 15 | 15 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 |
| size | 1 | 1 | 4 | 12 | 20 | 30 | 2 | 2 | 30 | 60 | 2 | 2 | 2 | 8 | 40 | 12 | 20 | 2 | 2 | 8 | 8 | 24 | 24 | 4 | 4 | 4 | 4 | 4 | 20 | 20 | 4 | 4 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C8⋊C22 | S3×D4 | S3×D5 | D4×D5 | D8⋊S3 | C2×S3×D5 | D8⋊D5 | D10⋊D6 | D30.8D4 |
| kernel | D30.8D4 | D30.5C4 | C6.D20 | D12.D5 | C3×D4⋊D5 | C5×D4⋊S3 | C20⋊D6 | D4⋊2D15 | D4⋊D5 | Dic15 | D30 | D4⋊S3 | C5⋊2C8 | D20 | C5×D4 | C3⋊C8 | D12 | C3×D4 | C15 | C10 | D4 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of D30.8D4 ►in GL6(𝔽241)
| 189 | 1 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 1 | 189 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 194 | 47 | 194 | 47 |
| 0 | 0 | 194 | 147 | 194 | 147 |
| 0 | 0 | 194 | 47 | 47 | 194 |
| 0 | 0 | 194 | 147 | 47 | 94 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 47 | 194 | 194 | 47 |
| 0 | 0 | 147 | 194 | 94 | 47 |
| 0 | 0 | 47 | 194 | 47 | 194 |
| 0 | 0 | 147 | 194 | 147 | 194 |
G:=sub<GL(6,GF(241))| [189,240,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,189,240,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,194,194,194,194,0,0,47,147,47,147,0,0,194,194,47,47,0,0,47,147,194,94],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,47,147,47,147,0,0,194,194,194,194,0,0,194,94,47,147,0,0,47,47,194,194] >;
D30.8D4 in GAP, Magma, Sage, TeX
D_{30}._8D_4 % in TeX
G:=Group("D30.8D4"); // GroupNames label
G:=SmallGroup(480,558);
// by ID
G=gap.SmallGroup(480,558);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,303,675,346,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^6=1,d^2=a^5,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=d*b*d^-1=a^15*b,d*c*d^-1=a^5*c^-1>;
// generators/relations