metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊D15, C15⋊7D8, D60⋊2C2, C20.9D6, C4.1D30, C30.34D4, C12.9D10, C60.1C22, C5⋊3(D4⋊S3), C3⋊3(D4⋊D5), (C5×D4)⋊1S3, (C3×D4)⋊1D5, C15⋊3C8⋊1C2, (D4×C15)⋊1C2, C6.16(C5⋊D4), C2.4(C15⋊7D4), C10.16(C3⋊D4), SmallGroup(240,76)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊D15
G = < a,b,c,d | a4=b2=c15=d2=1, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=c-1 >
Smallest permutation representation of D4⋊D15
►On 120 points
Generators in S120
(1 54 24 43)(2 55 25 44)(3 56 26 45)(4 57 27 31)(5 58 28 32)(6 59 29 33)(7 60 30 34)(8 46 16 35)(9 47 17 36)(10 48 18 37)(11 49 19 38)(12 50 20 39)(13 51 21 40)(14 52 22 41)(15 53 23 42)(61 95 80 114)(62 96 81 115)(63 97 82 116)(64 98 83 117)(65 99 84 118)(66 100 85 119)(67 101 86 120)(68 102 87 106)(69 103 88 107)(70 104 89 108)(71 105 90 109)(72 91 76 110)(73 92 77 111)(74 93 78 112)(75 94 79 113)
(1 114)(2 115)(3 116)(4 117)(5 118)(6 119)(7 120)(8 106)(9 107)(10 108)(11 109)(12 110)(13 111)(14 112)(15 113)(16 102)(17 103)(18 104)(19 105)(20 91)(21 92)(22 93)(23 94)(24 95)(25 96)(26 97)(27 98)(28 99)(29 100)(30 101)(31 64)(32 65)(33 66)(34 67)(35 68)(36 69)(37 70)(38 71)(39 72)(40 73)(41 74)(42 75)(43 61)(44 62)(45 63)(46 87)(47 88)(48 89)(49 90)(50 76)(51 77)(52 78)(53 79)(54 80)(55 81)(56 82)(57 83)(58 84)(59 85)(60 86)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 24)(31 50)(32 49)(33 48)(34 47)(35 46)(36 60)(37 59)(38 58)(39 57)(40 56)(41 55)(42 54)(43 53)(44 52)(45 51)(61 94)(62 93)(63 92)(64 91)(65 105)(66 104)(67 103)(68 102)(69 101)(70 100)(71 99)(72 98)(73 97)(74 96)(75 95)(76 117)(77 116)(78 115)(79 114)(80 113)(81 112)(82 111)(83 110)(84 109)(85 108)(86 107)(87 106)(88 120)(89 119)(90 118)
G:=sub<Sym(120)| (1,54,24,43)(2,55,25,44)(3,56,26,45)(4,57,27,31)(5,58,28,32)(6,59,29,33)(7,60,30,34)(8,46,16,35)(9,47,17,36)(10,48,18,37)(11,49,19,38)(12,50,20,39)(13,51,21,40)(14,52,22,41)(15,53,23,42)(61,95,80,114)(62,96,81,115)(63,97,82,116)(64,98,83,117)(65,99,84,118)(66,100,85,119)(67,101,86,120)(68,102,87,106)(69,103,88,107)(70,104,89,108)(71,105,90,109)(72,91,76,110)(73,92,77,111)(74,93,78,112)(75,94,79,113), (1,114)(2,115)(3,116)(4,117)(5,118)(6,119)(7,120)(8,106)(9,107)(10,108)(11,109)(12,110)(13,111)(14,112)(15,113)(16,102)(17,103)(18,104)(19,105)(20,91)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,100)(30,101)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73)(41,74)(42,75)(43,61)(44,62)(45,63)(46,87)(47,88)(48,89)(49,90)(50,76)(51,77)(52,78)(53,79)(54,80)(55,81)(56,82)(57,83)(58,84)(59,85)(60,86), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(31,50)(32,49)(33,48)(34,47)(35,46)(36,60)(37,59)(38,58)(39,57)(40,56)(41,55)(42,54)(43,53)(44,52)(45,51)(61,94)(62,93)(63,92)(64,91)(65,105)(66,104)(67,103)(68,102)(69,101)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,117)(77,116)(78,115)(79,114)(80,113)(81,112)(82,111)(83,110)(84,109)(85,108)(86,107)(87,106)(88,120)(89,119)(90,118)>;
G:=Group( (1,54,24,43)(2,55,25,44)(3,56,26,45)(4,57,27,31)(5,58,28,32)(6,59,29,33)(7,60,30,34)(8,46,16,35)(9,47,17,36)(10,48,18,37)(11,49,19,38)(12,50,20,39)(13,51,21,40)(14,52,22,41)(15,53,23,42)(61,95,80,114)(62,96,81,115)(63,97,82,116)(64,98,83,117)(65,99,84,118)(66,100,85,119)(67,101,86,120)(68,102,87,106)(69,103,88,107)(70,104,89,108)(71,105,90,109)(72,91,76,110)(73,92,77,111)(74,93,78,112)(75,94,79,113), (1,114)(2,115)(3,116)(4,117)(5,118)(6,119)(7,120)(8,106)(9,107)(10,108)(11,109)(12,110)(13,111)(14,112)(15,113)(16,102)(17,103)(18,104)(19,105)(20,91)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,100)(30,101)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73)(41,74)(42,75)(43,61)(44,62)(45,63)(46,87)(47,88)(48,89)(49,90)(50,76)(51,77)(52,78)(53,79)(54,80)(55,81)(56,82)(57,83)(58,84)(59,85)(60,86), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(31,50)(32,49)(33,48)(34,47)(35,46)(36,60)(37,59)(38,58)(39,57)(40,56)(41,55)(42,54)(43,53)(44,52)(45,51)(61,94)(62,93)(63,92)(64,91)(65,105)(66,104)(67,103)(68,102)(69,101)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,117)(77,116)(78,115)(79,114)(80,113)(81,112)(82,111)(83,110)(84,109)(85,108)(86,107)(87,106)(88,120)(89,119)(90,118) );
G=PermutationGroup([[(1,54,24,43),(2,55,25,44),(3,56,26,45),(4,57,27,31),(5,58,28,32),(6,59,29,33),(7,60,30,34),(8,46,16,35),(9,47,17,36),(10,48,18,37),(11,49,19,38),(12,50,20,39),(13,51,21,40),(14,52,22,41),(15,53,23,42),(61,95,80,114),(62,96,81,115),(63,97,82,116),(64,98,83,117),(65,99,84,118),(66,100,85,119),(67,101,86,120),(68,102,87,106),(69,103,88,107),(70,104,89,108),(71,105,90,109),(72,91,76,110),(73,92,77,111),(74,93,78,112),(75,94,79,113)], [(1,114),(2,115),(3,116),(4,117),(5,118),(6,119),(7,120),(8,106),(9,107),(10,108),(11,109),(12,110),(13,111),(14,112),(15,113),(16,102),(17,103),(18,104),(19,105),(20,91),(21,92),(22,93),(23,94),(24,95),(25,96),(26,97),(27,98),(28,99),(29,100),(30,101),(31,64),(32,65),(33,66),(34,67),(35,68),(36,69),(37,70),(38,71),(39,72),(40,73),(41,74),(42,75),(43,61),(44,62),(45,63),(46,87),(47,88),(48,89),(49,90),(50,76),(51,77),(52,78),(53,79),(54,80),(55,81),(56,82),(57,83),(58,84),(59,85),(60,86)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(17,30),(18,29),(19,28),(20,27),(21,26),(22,25),(23,24),(31,50),(32,49),(33,48),(34,47),(35,46),(36,60),(37,59),(38,58),(39,57),(40,56),(41,55),(42,54),(43,53),(44,52),(45,51),(61,94),(62,93),(63,92),(64,91),(65,105),(66,104),(67,103),(68,102),(69,101),(70,100),(71,99),(72,98),(73,97),(74,96),(75,95),(76,117),(77,116),(78,115),(79,114),(80,113),(81,112),(82,111),(83,110),(84,109),(85,108),(86,107),(87,106),(88,120),(89,119),(90,118)]])
D4⋊D15 is a maximal subgroup of
D5×D4⋊S3 Dic10⋊3D6 S3×D4⋊D5 D60.C22 D20.9D6 C60.16C23 C60.19C23 D12.9D10 D8×D15 D8⋊D15 Q8⋊3D30 D4.5D30 D4.D30 D4⋊D30 D4.8D30
D4⋊D15 is a maximal quotient of
C60.1Q8 D60⋊9C4 C15⋊7D16 D8.D15 C8.6D30 C15⋊7Q32 D4⋊Dic15
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 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 | 2 | 3 | 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 | 60 | 2 | 2 | 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 | D8 | D10 | C3⋊D4 | D15 | C5⋊D4 | D30 | C15⋊7D4 | D4⋊S3 | D4⋊D5 | D4⋊D15 |
| kernel | D4⋊D15 | C15⋊3C8 | D60 | 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 GL4(𝔽241) generated by
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 1 | 192 |
| 0 | 0 | 123 | 240 |
| 76 | 49 | 0 | 0 |
| 192 | 165 | 0 | 0 |
| 0 | 0 | 0 | 184 |
| 0 | 0 | 93 | 0 |
| 148 | 80 | 0 | 0 |
| 161 | 131 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 93 | 161 | 0 | 0 |
| 84 | 148 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 123 | 240 |
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,1,123,0,0,192,240],[76,192,0,0,49,165,0,0,0,0,0,93,0,0,184,0],[148,161,0,0,80,131,0,0,0,0,1,0,0,0,0,1],[93,84,0,0,161,148,0,0,0,0,1,123,0,0,0,240] >;
D4⋊D15 in GAP, Magma, Sage, TeX
D_4\rtimes D_{15} % in TeX
G:=Group("D4:D15"); // GroupNames label
G:=SmallGroup(240,76);
// by ID
G=gap.SmallGroup(240,76);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,73,218,116,50,964,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^15=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations
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