metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4.11D60, C12.11D20, C20.11D12, C60.203D4, M4(2)⋊3D15, (C2×C4).1D30, (C2×D60).7C2, (C2×C20).69D6, (C2×C12).70D10, C15⋊7(C4.D4), (C5×M4(2))⋊9S3, (C3×M4(2))⋊9D5, C60.7C4⋊14C2, C22.4(C4×D15), C10.34(D6⋊C4), C4.21(C15⋊7D4), C5⋊3(C12.46D4), C3⋊2(C20.46D4), (C2×C60).55C22, (C22×D15).1C4, C2.9(D30⋊3C4), C20.100(C3⋊D4), C12.100(C5⋊D4), C30.76(C22⋊C4), (C15×M4(2))⋊19C2, C6.19(D10⋊C4), (C2×C6).5(C4×D5), (C2×C10).28(C4×S3), (C2×C30).65(C2×C4), SmallGroup(480,183)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)⋊D15
G = < a,b,c,d | a8=b2=c15=d2=1, bab=a5, ac=ca, dad=ab, bc=cb, bd=db, dcd=c-1 >
Subgroups: 788 in 92 conjugacy classes, 33 normal (31 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, D4, C23, D5, C10, C10, C12, D6, C2×C6, C15, M4(2), M4(2), C2×D4, C20, D10, C2×C10, C3⋊C8, C24, D12, C2×C12, C22×S3, D15, C30, C30, C4.D4, C5⋊2C8, C40, D20, C2×C20, C22×D5, C4.Dic3, C3×M4(2), C2×D12, C60, D30, C2×C30, C4.Dic5, C5×M4(2), C2×D20, C12.46D4, C15⋊3C8, C120, D60, C2×C60, C22×D15, C20.46D4, C60.7C4, C15×M4(2), C2×D60, M4(2)⋊D15
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, D6, C22⋊C4, D10, C4×S3, D12, C3⋊D4, D15, C4.D4, C4×D5, D20, C5⋊D4, D6⋊C4, D30, D10⋊C4, C12.46D4, C4×D15, D60, C15⋊7D4, C20.46D4, D30⋊3C4, M4(2)⋊D15
►On 120 points
Generators in S120
(1 91 39 80 16 107 47 73)(2 92 40 81 17 108 48 74)(3 93 41 82 18 109 49 75)(4 94 42 83 19 110 50 61)(5 95 43 84 20 111 51 62)(6 96 44 85 21 112 52 63)(7 97 45 86 22 113 53 64)(8 98 31 87 23 114 54 65)(9 99 32 88 24 115 55 66)(10 100 33 89 25 116 56 67)(11 101 34 90 26 117 57 68)(12 102 35 76 27 118 58 69)(13 103 36 77 28 119 59 70)(14 104 37 78 29 120 60 71)(15 105 38 79 30 106 46 72)
(61 83)(62 84)(63 85)(64 86)(65 87)(66 88)(67 89)(68 90)(69 76)(70 77)(71 78)(72 79)(73 80)(74 81)(75 82)(91 107)(92 108)(93 109)(94 110)(95 111)(96 112)(97 113)(98 114)(99 115)(100 116)(101 117)(102 118)(103 119)(104 120)(105 106)
(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)(16 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(31 54)(32 53)(33 52)(34 51)(35 50)(36 49)(37 48)(38 47)(39 46)(40 60)(41 59)(42 58)(43 57)(44 56)(45 55)(61 69)(62 68)(63 67)(64 66)(70 75)(71 74)(72 73)(76 83)(77 82)(78 81)(79 80)(84 90)(85 89)(86 88)(91 106)(92 120)(93 119)(94 118)(95 117)(96 116)(97 115)(98 114)(99 113)(100 112)(101 111)(102 110)(103 109)(104 108)(105 107)
G:=sub<Sym(120)| (1,91,39,80,16,107,47,73)(2,92,40,81,17,108,48,74)(3,93,41,82,18,109,49,75)(4,94,42,83,19,110,50,61)(5,95,43,84,20,111,51,62)(6,96,44,85,21,112,52,63)(7,97,45,86,22,113,53,64)(8,98,31,87,23,114,54,65)(9,99,32,88,24,115,55,66)(10,100,33,89,25,116,56,67)(11,101,34,90,26,117,57,68)(12,102,35,76,27,118,58,69)(13,103,36,77,28,119,59,70)(14,104,37,78,29,120,60,71)(15,105,38,79,30,106,46,72), (61,83)(62,84)(63,85)(64,86)(65,87)(66,88)(67,89)(68,90)(69,76)(70,77)(71,78)(72,79)(73,80)(74,81)(75,82)(91,107)(92,108)(93,109)(94,110)(95,111)(96,112)(97,113)(98,114)(99,115)(100,116)(101,117)(102,118)(103,119)(104,120)(105,106), (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)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(31,54)(32,53)(33,52)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,60)(41,59)(42,58)(43,57)(44,56)(45,55)(61,69)(62,68)(63,67)(64,66)(70,75)(71,74)(72,73)(76,83)(77,82)(78,81)(79,80)(84,90)(85,89)(86,88)(91,106)(92,120)(93,119)(94,118)(95,117)(96,116)(97,115)(98,114)(99,113)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)>;
G:=Group( (1,91,39,80,16,107,47,73)(2,92,40,81,17,108,48,74)(3,93,41,82,18,109,49,75)(4,94,42,83,19,110,50,61)(5,95,43,84,20,111,51,62)(6,96,44,85,21,112,52,63)(7,97,45,86,22,113,53,64)(8,98,31,87,23,114,54,65)(9,99,32,88,24,115,55,66)(10,100,33,89,25,116,56,67)(11,101,34,90,26,117,57,68)(12,102,35,76,27,118,58,69)(13,103,36,77,28,119,59,70)(14,104,37,78,29,120,60,71)(15,105,38,79,30,106,46,72), (61,83)(62,84)(63,85)(64,86)(65,87)(66,88)(67,89)(68,90)(69,76)(70,77)(71,78)(72,79)(73,80)(74,81)(75,82)(91,107)(92,108)(93,109)(94,110)(95,111)(96,112)(97,113)(98,114)(99,115)(100,116)(101,117)(102,118)(103,119)(104,120)(105,106), (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)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(31,54)(32,53)(33,52)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,60)(41,59)(42,58)(43,57)(44,56)(45,55)(61,69)(62,68)(63,67)(64,66)(70,75)(71,74)(72,73)(76,83)(77,82)(78,81)(79,80)(84,90)(85,89)(86,88)(91,106)(92,120)(93,119)(94,118)(95,117)(96,116)(97,115)(98,114)(99,113)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107) );
G=PermutationGroup([[(1,91,39,80,16,107,47,73),(2,92,40,81,17,108,48,74),(3,93,41,82,18,109,49,75),(4,94,42,83,19,110,50,61),(5,95,43,84,20,111,51,62),(6,96,44,85,21,112,52,63),(7,97,45,86,22,113,53,64),(8,98,31,87,23,114,54,65),(9,99,32,88,24,115,55,66),(10,100,33,89,25,116,56,67),(11,101,34,90,26,117,57,68),(12,102,35,76,27,118,58,69),(13,103,36,77,28,119,59,70),(14,104,37,78,29,120,60,71),(15,105,38,79,30,106,46,72)], [(61,83),(62,84),(63,85),(64,86),(65,87),(66,88),(67,89),(68,90),(69,76),(70,77),(71,78),(72,79),(73,80),(74,81),(75,82),(91,107),(92,108),(93,109),(94,110),(95,111),(96,112),(97,113),(98,114),(99,115),(100,116),(101,117),(102,118),(103,119),(104,120),(105,106)], [(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),(16,30),(17,29),(18,28),(19,27),(20,26),(21,25),(22,24),(31,54),(32,53),(33,52),(34,51),(35,50),(36,49),(37,48),(38,47),(39,46),(40,60),(41,59),(42,58),(43,57),(44,56),(45,55),(61,69),(62,68),(63,67),(64,66),(70,75),(71,74),(72,73),(76,83),(77,82),(78,81),(79,80),(84,90),(85,89),(86,88),(91,106),(92,120),(93,119),(94,118),(95,117),(96,116),(97,115),(98,114),(99,113),(100,112),(101,111),(102,110),(103,109),(104,108),(105,107)]])
81 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 5A | 5B | 6A | 6B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | 40A | ··· | 40H | 60A | ··· | 60H | 60I | 60J | 60K | 60L | 120A | ··· | 120P |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 60 | 60 | 2 | 2 | 4 | 4 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
81 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D5 | D6 | D10 | D12 | C3⋊D4 | C4×S3 | D15 | D20 | C5⋊D4 | C4×D5 | D30 | D60 | C15⋊7D4 | C4×D15 | C4.D4 | C12.46D4 | C20.46D4 | M4(2)⋊D15 |
| kernel | M4(2)⋊D15 | C60.7C4 | C15×M4(2) | C2×D60 | C22×D15 | C5×M4(2) | C60 | C3×M4(2) | C2×C20 | C2×C12 | C20 | C20 | C2×C10 | M4(2) | C12 | C12 | C2×C6 | C2×C4 | C4 | C4 | C22 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of M4(2)⋊D15 ►in GL4(𝔽241) generated by
| 0 | 0 | 240 | 1 |
| 51 | 240 | 239 | 51 |
| 83 | 178 | 1 | 0 |
| 124 | 181 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 234 | 123 | 240 | 0 |
| 234 | 123 | 0 | 240 |
| 225 | 110 | 0 | 0 |
| 195 | 30 | 0 | 0 |
| 110 | 0 | 94 | 110 |
| 177 | 131 | 131 | 161 |
| 63 | 94 | 0 | 0 |
| 127 | 178 | 0 | 0 |
| 110 | 0 | 94 | 110 |
| 84 | 147 | 84 | 147 |
G:=sub<GL(4,GF(241))| [0,51,83,124,0,240,178,181,240,239,1,1,1,51,0,0],[1,0,234,234,0,1,123,123,0,0,240,0,0,0,0,240],[225,195,110,177,110,30,0,131,0,0,94,131,0,0,110,161],[63,127,110,84,94,178,0,147,0,0,94,84,0,0,110,147] >;
M4(2)⋊D15 in GAP, Magma, Sage, TeX
M_4(2)\rtimes D_{15} % in TeX
G:=Group("M4(2):D15"); // GroupNames label
G:=SmallGroup(480,183);
// by ID
G=gap.SmallGroup(480,183);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,36,422,100,346,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^15=d^2=1,b*a*b=a^5,a*c=c*a,d*a*d=a*b,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations