metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.D15, C4.2D30, C20.10D6, C30.35D4, C15⋊10SD16, Dic30⋊2C2, C12.10D10, C60.2C22, C15⋊3C8⋊2C2, C3⋊3(D4.D5), C5⋊3(D4.S3), (C3×D4).1D5, (C5×D4).1S3, (D4×C15).1C2, C6.17(C5⋊D4), C2.5(C15⋊7D4), C10.17(C3⋊D4), SmallGroup(240,77)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.D15
G = < a,b,c,d | a4=b2=c15=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >
Smallest permutation representation of D4.D15
►On 120 points
Generators in S120
(1 50 20 43)(2 51 21 44)(3 52 22 45)(4 53 23 31)(5 54 24 32)(6 55 25 33)(7 56 26 34)(8 57 27 35)(9 58 28 36)(10 59 29 37)(11 60 30 38)(12 46 16 39)(13 47 17 40)(14 48 18 41)(15 49 19 42)(61 99 81 113)(62 100 82 114)(63 101 83 115)(64 102 84 116)(65 103 85 117)(66 104 86 118)(67 105 87 119)(68 91 88 120)(69 92 89 106)(70 93 90 107)(71 94 76 108)(72 95 77 109)(73 96 78 110)(74 97 79 111)(75 98 80 112)
(1 43)(2 44)(3 45)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(61 81)(62 82)(63 83)(64 84)(65 85)(66 86)(67 87)(68 88)(69 89)(70 90)(71 76)(72 77)(73 78)(74 79)(75 80)
(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 87 20 67)(2 86 21 66)(3 85 22 65)(4 84 23 64)(5 83 24 63)(6 82 25 62)(7 81 26 61)(8 80 27 75)(9 79 28 74)(10 78 29 73)(11 77 30 72)(12 76 16 71)(13 90 17 70)(14 89 18 69)(15 88 19 68)(31 116 53 102)(32 115 54 101)(33 114 55 100)(34 113 56 99)(35 112 57 98)(36 111 58 97)(37 110 59 96)(38 109 60 95)(39 108 46 94)(40 107 47 93)(41 106 48 92)(42 120 49 91)(43 119 50 105)(44 118 51 104)(45 117 52 103)
G:=sub<Sym(120)| (1,50,20,43)(2,51,21,44)(3,52,22,45)(4,53,23,31)(5,54,24,32)(6,55,25,33)(7,56,26,34)(8,57,27,35)(9,58,28,36)(10,59,29,37)(11,60,30,38)(12,46,16,39)(13,47,17,40)(14,48,18,41)(15,49,19,42)(61,99,81,113)(62,100,82,114)(63,101,83,115)(64,102,84,116)(65,103,85,117)(66,104,86,118)(67,105,87,119)(68,91,88,120)(69,92,89,106)(70,93,90,107)(71,94,76,108)(72,95,77,109)(73,96,78,110)(74,97,79,111)(75,98,80,112), (1,43)(2,44)(3,45)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,76)(72,77)(73,78)(74,79)(75,80), (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), (1,87,20,67)(2,86,21,66)(3,85,22,65)(4,84,23,64)(5,83,24,63)(6,82,25,62)(7,81,26,61)(8,80,27,75)(9,79,28,74)(10,78,29,73)(11,77,30,72)(12,76,16,71)(13,90,17,70)(14,89,18,69)(15,88,19,68)(31,116,53,102)(32,115,54,101)(33,114,55,100)(34,113,56,99)(35,112,57,98)(36,111,58,97)(37,110,59,96)(38,109,60,95)(39,108,46,94)(40,107,47,93)(41,106,48,92)(42,120,49,91)(43,119,50,105)(44,118,51,104)(45,117,52,103)>;
G:=Group( (1,50,20,43)(2,51,21,44)(3,52,22,45)(4,53,23,31)(5,54,24,32)(6,55,25,33)(7,56,26,34)(8,57,27,35)(9,58,28,36)(10,59,29,37)(11,60,30,38)(12,46,16,39)(13,47,17,40)(14,48,18,41)(15,49,19,42)(61,99,81,113)(62,100,82,114)(63,101,83,115)(64,102,84,116)(65,103,85,117)(66,104,86,118)(67,105,87,119)(68,91,88,120)(69,92,89,106)(70,93,90,107)(71,94,76,108)(72,95,77,109)(73,96,78,110)(74,97,79,111)(75,98,80,112), (1,43)(2,44)(3,45)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,76)(72,77)(73,78)(74,79)(75,80), (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), (1,87,20,67)(2,86,21,66)(3,85,22,65)(4,84,23,64)(5,83,24,63)(6,82,25,62)(7,81,26,61)(8,80,27,75)(9,79,28,74)(10,78,29,73)(11,77,30,72)(12,76,16,71)(13,90,17,70)(14,89,18,69)(15,88,19,68)(31,116,53,102)(32,115,54,101)(33,114,55,100)(34,113,56,99)(35,112,57,98)(36,111,58,97)(37,110,59,96)(38,109,60,95)(39,108,46,94)(40,107,47,93)(41,106,48,92)(42,120,49,91)(43,119,50,105)(44,118,51,104)(45,117,52,103) );
G=PermutationGroup([[(1,50,20,43),(2,51,21,44),(3,52,22,45),(4,53,23,31),(5,54,24,32),(6,55,25,33),(7,56,26,34),(8,57,27,35),(9,58,28,36),(10,59,29,37),(11,60,30,38),(12,46,16,39),(13,47,17,40),(14,48,18,41),(15,49,19,42),(61,99,81,113),(62,100,82,114),(63,101,83,115),(64,102,84,116),(65,103,85,117),(66,104,86,118),(67,105,87,119),(68,91,88,120),(69,92,89,106),(70,93,90,107),(71,94,76,108),(72,95,77,109),(73,96,78,110),(74,97,79,111),(75,98,80,112)], [(1,43),(2,44),(3,45),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,37),(11,38),(12,39),(13,40),(14,41),(15,42),(16,46),(17,47),(18,48),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(61,81),(62,82),(63,83),(64,84),(65,85),(66,86),(67,87),(68,88),(69,89),(70,90),(71,76),(72,77),(73,78),(74,79),(75,80)], [(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)], [(1,87,20,67),(2,86,21,66),(3,85,22,65),(4,84,23,64),(5,83,24,63),(6,82,25,62),(7,81,26,61),(8,80,27,75),(9,79,28,74),(10,78,29,73),(11,77,30,72),(12,76,16,71),(13,90,17,70),(14,89,18,69),(15,88,19,68),(31,116,53,102),(32,115,54,101),(33,114,55,100),(34,113,56,99),(35,112,57,98),(36,111,58,97),(37,110,59,96),(38,109,60,95),(39,108,46,94),(40,107,47,93),(41,106,48,92),(42,120,49,91),(43,119,50,105),(44,118,51,104),(45,117,52,103)]])
D4.D15 is a maximal subgroup of
D5×D4.S3 C60.8C23 S3×D4.D5 C60.10C23 D12⋊10D10 D12.24D10 D20.24D6 D20⋊10D6 D8⋊D15 D8⋊3D15 SD16×D15 SD16⋊D15 D4.D30 D4.8D30 D4.9D30
D4.D15 is a maximal quotient of
C60.2Q8 Dic30⋊9C4 D4⋊Dic15
42 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 12 | 15A | 15B | 15C | 15D | 20A | 20B | 30A | 30B | 30C | 30D | 30E | ··· | 30L | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 4 | 2 | 2 | 60 | 2 | 2 | 2 | 4 | 4 | 30 | 30 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | - | - | ||||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | SD16 | D10 | C3⋊D4 | D15 | C5⋊D4 | D30 | C15⋊7D4 | D4.S3 | D4.D5 | D4.D15 |
| kernel | D4.D15 | C15⋊3C8 | Dic30 | D4×C15 | C5×D4 | C30 | C3×D4 | C20 | C15 | C12 | C10 | D4 | C6 | C4 | C2 | C5 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 1 | 2 | 4 |
Matrix representation of D4.D15 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 36 |
| 0 | 0 | 0 | 0 | 174 | 240 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 171 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 36 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 240 | 51 | 0 | 0 | 0 | 0 |
| 190 | 190 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 121 | 0 | 0 |
| 0 | 0 | 0 | 225 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 190 | 190 | 0 | 0 | 0 | 0 |
| 240 | 51 | 0 | 0 | 0 | 0 |
| 0 | 0 | 119 | 11 | 0 | 0 |
| 0 | 0 | 93 | 122 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 202 |
| 0 | 0 | 0 | 0 | 68 | 0 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,174,0,0,0,0,36,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,171,1,0,0,0,0,0,0,1,0,0,0,0,0,36,240],[240,190,0,0,0,0,51,190,0,0,0,0,0,0,15,0,0,0,0,0,121,225,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[190,240,0,0,0,0,190,51,0,0,0,0,0,0,119,93,0,0,0,0,11,122,0,0,0,0,0,0,0,68,0,0,0,0,202,0] >;
D4.D15 in GAP, Magma, Sage, TeX
D_4.D_{15} % in TeX
G:=Group("D4.D15"); // GroupNames label
G:=SmallGroup(240,77);
// by ID
G=gap.SmallGroup(240,77);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,73,218,116,50,964,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^15=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations
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