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 58 20 43)(2 59 21 44)(3 60 22 45)(4 46 23 31)(5 47 24 32)(6 48 25 33)(7 49 26 34)(8 50 27 35)(9 51 28 36)(10 52 29 37)(11 53 30 38)(12 54 16 39)(13 55 17 40)(14 56 18 41)(15 57 19 42)(61 101 90 108)(62 102 76 109)(63 103 77 110)(64 104 78 111)(65 105 79 112)(66 91 80 113)(67 92 81 114)(68 93 82 115)(69 94 83 116)(70 95 84 117)(71 96 85 118)(72 97 86 119)(73 98 87 120)(74 99 88 106)(75 100 89 107)
(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 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 46)(24 47)(25 48)(26 49)(27 50)(28 51)(29 52)(30 53)(61 90)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(71 85)(72 86)(73 87)(74 88)(75 89)
(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 86 20 72)(2 85 21 71)(3 84 22 70)(4 83 23 69)(5 82 24 68)(6 81 25 67)(7 80 26 66)(8 79 27 65)(9 78 28 64)(10 77 29 63)(11 76 30 62)(12 90 16 61)(13 89 17 75)(14 88 18 74)(15 87 19 73)(31 116 46 94)(32 115 47 93)(33 114 48 92)(34 113 49 91)(35 112 50 105)(36 111 51 104)(37 110 52 103)(38 109 53 102)(39 108 54 101)(40 107 55 100)(41 106 56 99)(42 120 57 98)(43 119 58 97)(44 118 59 96)(45 117 60 95)
G:=sub<Sym(120)| (1,58,20,43)(2,59,21,44)(3,60,22,45)(4,46,23,31)(5,47,24,32)(6,48,25,33)(7,49,26,34)(8,50,27,35)(9,51,28,36)(10,52,29,37)(11,53,30,38)(12,54,16,39)(13,55,17,40)(14,56,18,41)(15,57,19,42)(61,101,90,108)(62,102,76,109)(63,103,77,110)(64,104,78,111)(65,105,79,112)(66,91,80,113)(67,92,81,114)(68,93,82,115)(69,94,83,116)(70,95,84,117)(71,96,85,118)(72,97,86,119)(73,98,87,120)(74,99,88,106)(75,100,89,107), (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)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,46)(24,47)(25,48)(26,49)(27,50)(28,51)(29,52)(30,53)(61,90)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(71,85)(72,86)(73,87)(74,88)(75,89), (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), (1,86,20,72)(2,85,21,71)(3,84,22,70)(4,83,23,69)(5,82,24,68)(6,81,25,67)(7,80,26,66)(8,79,27,65)(9,78,28,64)(10,77,29,63)(11,76,30,62)(12,90,16,61)(13,89,17,75)(14,88,18,74)(15,87,19,73)(31,116,46,94)(32,115,47,93)(33,114,48,92)(34,113,49,91)(35,112,50,105)(36,111,51,104)(37,110,52,103)(38,109,53,102)(39,108,54,101)(40,107,55,100)(41,106,56,99)(42,120,57,98)(43,119,58,97)(44,118,59,96)(45,117,60,95)>;
G:=Group( (1,58,20,43)(2,59,21,44)(3,60,22,45)(4,46,23,31)(5,47,24,32)(6,48,25,33)(7,49,26,34)(8,50,27,35)(9,51,28,36)(10,52,29,37)(11,53,30,38)(12,54,16,39)(13,55,17,40)(14,56,18,41)(15,57,19,42)(61,101,90,108)(62,102,76,109)(63,103,77,110)(64,104,78,111)(65,105,79,112)(66,91,80,113)(67,92,81,114)(68,93,82,115)(69,94,83,116)(70,95,84,117)(71,96,85,118)(72,97,86,119)(73,98,87,120)(74,99,88,106)(75,100,89,107), (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)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,46)(24,47)(25,48)(26,49)(27,50)(28,51)(29,52)(30,53)(61,90)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(71,85)(72,86)(73,87)(74,88)(75,89), (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), (1,86,20,72)(2,85,21,71)(3,84,22,70)(4,83,23,69)(5,82,24,68)(6,81,25,67)(7,80,26,66)(8,79,27,65)(9,78,28,64)(10,77,29,63)(11,76,30,62)(12,90,16,61)(13,89,17,75)(14,88,18,74)(15,87,19,73)(31,116,46,94)(32,115,47,93)(33,114,48,92)(34,113,49,91)(35,112,50,105)(36,111,51,104)(37,110,52,103)(38,109,53,102)(39,108,54,101)(40,107,55,100)(41,106,56,99)(42,120,57,98)(43,119,58,97)(44,118,59,96)(45,117,60,95) );
G=PermutationGroup([(1,58,20,43),(2,59,21,44),(3,60,22,45),(4,46,23,31),(5,47,24,32),(6,48,25,33),(7,49,26,34),(8,50,27,35),(9,51,28,36),(10,52,29,37),(11,53,30,38),(12,54,16,39),(13,55,17,40),(14,56,18,41),(15,57,19,42),(61,101,90,108),(62,102,76,109),(63,103,77,110),(64,104,78,111),(65,105,79,112),(66,91,80,113),(67,92,81,114),(68,93,82,115),(69,94,83,116),(70,95,84,117),(71,96,85,118),(72,97,86,119),(73,98,87,120),(74,99,88,106),(75,100,89,107)], [(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),(16,54),(17,55),(18,56),(19,57),(20,58),(21,59),(22,60),(23,46),(24,47),(25,48),(26,49),(27,50),(28,51),(29,52),(30,53),(61,90),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(71,85),(72,86),(73,87),(74,88),(75,89)], [(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)], [(1,86,20,72),(2,85,21,71),(3,84,22,70),(4,83,23,69),(5,82,24,68),(6,81,25,67),(7,80,26,66),(8,79,27,65),(9,78,28,64),(10,77,29,63),(11,76,30,62),(12,90,16,61),(13,89,17,75),(14,88,18,74),(15,87,19,73),(31,116,46,94),(32,115,47,93),(33,114,48,92),(34,113,49,91),(35,112,50,105),(36,111,51,104),(37,110,52,103),(38,109,53,102),(39,108,54,101),(40,107,55,100),(41,106,56,99),(42,120,57,98),(43,119,58,97),(44,118,59,96),(45,117,60,95)])
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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