metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30.5C4, C20.34D6, C15⋊7M4(2), C12.34D10, C60.34C22, Dic15.5C4, C3⋊C8⋊5D5, C5⋊2C8⋊5S3, C6.2(C4×D5), C5⋊3(C8⋊S3), C3⋊1(C8⋊D5), C10.9(C4×S3), C4.27(S3×D5), C30.27(C2×C4), (C4×D15).5C2, C2.3(D30.C2), (C5×C3⋊C8)⋊7C2, (C3×C5⋊2C8)⋊7C2, SmallGroup(240,12)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30.5C4
G = < a,b,c | a30=b2=1, c4=a15, bab=a-1, cac-1=a19, cbc-1=a3b >
Smallest permutation representation of D30.5C4
►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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 60)(32 59)(33 58)(34 57)(35 56)(36 55)(37 54)(38 53)(39 52)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)(76 90)(77 89)(78 88)(79 87)(80 86)(81 85)(82 84)(91 95)(92 94)(96 120)(97 119)(98 118)(99 117)(100 116)(101 115)(102 114)(103 113)(104 112)(105 111)(106 110)(107 109)
(1 116 46 76 16 101 31 61)(2 105 47 65 17 120 32 80)(3 94 48 84 18 109 33 69)(4 113 49 73 19 98 34 88)(5 102 50 62 20 117 35 77)(6 91 51 81 21 106 36 66)(7 110 52 70 22 95 37 85)(8 99 53 89 23 114 38 74)(9 118 54 78 24 103 39 63)(10 107 55 67 25 92 40 82)(11 96 56 86 26 111 41 71)(12 115 57 75 27 100 42 90)(13 104 58 64 28 119 43 79)(14 93 59 83 29 108 44 68)(15 112 60 72 30 97 45 87)
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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(91,95)(92,94)(96,120)(97,119)(98,118)(99,117)(100,116)(101,115)(102,114)(103,113)(104,112)(105,111)(106,110)(107,109), (1,116,46,76,16,101,31,61)(2,105,47,65,17,120,32,80)(3,94,48,84,18,109,33,69)(4,113,49,73,19,98,34,88)(5,102,50,62,20,117,35,77)(6,91,51,81,21,106,36,66)(7,110,52,70,22,95,37,85)(8,99,53,89,23,114,38,74)(9,118,54,78,24,103,39,63)(10,107,55,67,25,92,40,82)(11,96,56,86,26,111,41,71)(12,115,57,75,27,100,42,90)(13,104,58,64,28,119,43,79)(14,93,59,83,29,108,44,68)(15,112,60,72,30,97,45,87)>;
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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(91,95)(92,94)(96,120)(97,119)(98,118)(99,117)(100,116)(101,115)(102,114)(103,113)(104,112)(105,111)(106,110)(107,109), (1,116,46,76,16,101,31,61)(2,105,47,65,17,120,32,80)(3,94,48,84,18,109,33,69)(4,113,49,73,19,98,34,88)(5,102,50,62,20,117,35,77)(6,91,51,81,21,106,36,66)(7,110,52,70,22,95,37,85)(8,99,53,89,23,114,38,74)(9,118,54,78,24,103,39,63)(10,107,55,67,25,92,40,82)(11,96,56,86,26,111,41,71)(12,115,57,75,27,100,42,90)(13,104,58,64,28,119,43,79)(14,93,59,83,29,108,44,68)(15,112,60,72,30,97,45,87) );
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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19),(13,18),(14,17),(15,16),(31,60),(32,59),(33,58),(34,57),(35,56),(36,55),(37,54),(38,53),(39,52),(40,51),(41,50),(42,49),(43,48),(44,47),(45,46),(61,75),(62,74),(63,73),(64,72),(65,71),(66,70),(67,69),(76,90),(77,89),(78,88),(79,87),(80,86),(81,85),(82,84),(91,95),(92,94),(96,120),(97,119),(98,118),(99,117),(100,116),(101,115),(102,114),(103,113),(104,112),(105,111),(106,110),(107,109)], [(1,116,46,76,16,101,31,61),(2,105,47,65,17,120,32,80),(3,94,48,84,18,109,33,69),(4,113,49,73,19,98,34,88),(5,102,50,62,20,117,35,77),(6,91,51,81,21,106,36,66),(7,110,52,70,22,95,37,85),(8,99,53,89,23,114,38,74),(9,118,54,78,24,103,39,63),(10,107,55,67,25,92,40,82),(11,96,56,86,26,111,41,71),(12,115,57,75,27,100,42,90),(13,104,58,64,28,119,43,79),(14,93,59,83,29,108,44,68),(15,112,60,72,30,97,45,87)]])
D30.5C4 is a maximal subgroup of
D5×C8⋊S3 S3×C8⋊D5 C40.54D6 C40.35D6 D60.5C4 D60.4C4 D15⋊4M4(2) D30.8D4 D30.9D4 Dic6⋊D10 D12⋊5D10 D60⋊C22 C60.C23 D20.17D6 D30.44D4
D30.5C4 is a maximal quotient of
C30.23C42 D30⋊4C8 C60.14Q8
42 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 5A | 5B | 6 | 8A | 8B | 8C | 8D | 10A | 10B | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30A | 30B | 40A | ··· | 40H | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 40 | ··· | 40 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 30 | 2 | 1 | 1 | 30 | 2 | 2 | 2 | 6 | 6 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 6 | ··· | 6 | 4 | 4 | 4 | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D5 | D6 | M4(2) | D10 | C4×S3 | C4×D5 | C8⋊S3 | C8⋊D5 | S3×D5 | D30.C2 | D30.5C4 |
| kernel | D30.5C4 | C5×C3⋊C8 | C3×C5⋊2C8 | C4×D15 | Dic15 | D30 | C5⋊2C8 | C3⋊C8 | C20 | C15 | C12 | C10 | C6 | C5 | C3 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 2 | 2 | 4 |
Matrix representation of D30.5C4 ►in GL4(𝔽241) generated by
| 1 | 1 | 0 | 0 |
| 188 | 189 | 0 | 0 |
| 0 | 0 | 240 | 1 |
| 0 | 0 | 240 | 0 |
| 0 | 190 | 0 | 0 |
| 189 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 240 | 1 |
| 229 | 99 | 0 | 0 |
| 99 | 12 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
G:=sub<GL(4,GF(241))| [1,188,0,0,1,189,0,0,0,0,240,240,0,0,1,0],[0,189,0,0,190,0,0,0,0,0,240,240,0,0,0,1],[229,99,0,0,99,12,0,0,0,0,240,0,0,0,0,240] >;
D30.5C4 in GAP, Magma, Sage, TeX
D_{30}._5C_4 % in TeX
G:=Group("D30.5C4"); // GroupNames label
G:=SmallGroup(240,12);
// by ID
G=gap.SmallGroup(240,12);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,31,50,490,6917]);
// Polycyclic
G:=Group<a,b,c|a^30=b^2=1,c^4=a^15,b*a*b=a^-1,c*a*c^-1=a^19,c*b*c^-1=a^3*b>;
// generators/relations
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