metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D15⋊4M4(2), C60.185C23, C3⋊C8⋊22D10, C5⋊2C8⋊22D6, C5⋊7(S3×M4(2)), C3⋊3(D5×M4(2)), C20.54(C4×S3), (C2×C20).82D6, C12.22(C4×D5), C4.Dic5⋊6S3, C4.Dic3⋊6D5, C60.131(C2×C4), D30.37(C2×C4), (C4×D15).10C4, (C2×C12).83D10, C15⋊17(C2×M4(2)), D15⋊2C8⋊13C2, D30.5C4⋊14C2, C30.107(C22×C4), (C2×C60).215C22, C20.182(C22×S3), C4.14(D30.C2), (C2×Dic15).22C4, Dic15.45(C2×C4), (C22×D15).13C4, (C4×D15).65C22, C12.182(C22×D5), C22.6(D30.C2), C6.41(C2×C4×D5), C10.74(S3×C2×C4), C4.155(C2×S3×D5), (C5×C3⋊C8)⋊22C22, (C2×C4×D15).18C2, (C2×C6).10(C4×D5), (C2×C10).33(C4×S3), (C2×C4).195(S3×D5), C2.6(C2×D30.C2), (C2×C30).104(C2×C4), (C3×C5⋊2C8)⋊22C22, (C5×C4.Dic3)⋊12C2, (C3×C4.Dic5)⋊12C2, SmallGroup(480,368)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D15⋊4M4(2)
G = < a,b,c,d | a15=b2=c8=d2=1, bab=a-1, cac-1=a11, ad=da, cbc-1=a10b, bd=db, dcd=c5 >
Subgroups: 668 in 136 conjugacy classes, 54 normal (40 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, M4(2), C22×C4, Dic5, C20, D10, C2×C10, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, D15, D15, C30, C30, C2×M4(2), C5⋊2C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, S3×C8, C8⋊S3, C4.Dic3, C3×M4(2), S3×C2×C4, Dic15, C60, D30, D30, C2×C30, C8×D5, C8⋊D5, C4.Dic5, C5×M4(2), C2×C4×D5, S3×M4(2), C5×C3⋊C8, C3×C5⋊2C8, C4×D15, C2×Dic15, C2×C60, C22×D15, D5×M4(2), D15⋊2C8, D30.5C4, C3×C4.Dic5, C5×C4.Dic3, C2×C4×D15, D15⋊4M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, M4(2), C22×C4, D10, C4×S3, C22×S3, C2×M4(2), C4×D5, C22×D5, S3×C2×C4, S3×D5, C2×C4×D5, S3×M4(2), D30.C2, C2×S3×D5, D5×M4(2), C2×D30.C2, D15⋊4M4(2)
►On 120 points
(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 23)(17 22)(18 21)(19 20)(24 30)(25 29)(26 28)(31 32)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(46 54)(47 53)(48 52)(49 51)(55 60)(56 59)(57 58)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(73 75)(76 83)(77 82)(78 81)(79 80)(84 90)(85 89)(86 88)(91 100)(92 99)(93 98)(94 97)(95 96)(101 105)(102 104)(106 117)(107 116)(108 115)(109 114)(110 113)(111 112)(118 120)
(1 112 32 67 20 96 58 80)(2 108 33 63 21 92 59 76)(3 119 34 74 22 103 60 87)(4 115 35 70 23 99 46 83)(5 111 36 66 24 95 47 79)(6 107 37 62 25 91 48 90)(7 118 38 73 26 102 49 86)(8 114 39 69 27 98 50 82)(9 110 40 65 28 94 51 78)(10 106 41 61 29 105 52 89)(11 117 42 72 30 101 53 85)(12 113 43 68 16 97 54 81)(13 109 44 64 17 93 55 77)(14 120 45 75 18 104 56 88)(15 116 31 71 19 100 57 84)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 16)(13 17)(14 18)(15 19)(31 57)(32 58)(33 59)(34 60)(35 46)(36 47)(37 48)(38 49)(39 50)(40 51)(41 52)(42 53)(43 54)(44 55)(45 56)
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,23)(17,22)(18,21)(19,20)(24,30)(25,29)(26,28)(31,32)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(46,54)(47,53)(48,52)(49,51)(55,60)(56,59)(57,58)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(73,75)(76,83)(77,82)(78,81)(79,80)(84,90)(85,89)(86,88)(91,100)(92,99)(93,98)(94,97)(95,96)(101,105)(102,104)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)(118,120), (1,112,32,67,20,96,58,80)(2,108,33,63,21,92,59,76)(3,119,34,74,22,103,60,87)(4,115,35,70,23,99,46,83)(5,111,36,66,24,95,47,79)(6,107,37,62,25,91,48,90)(7,118,38,73,26,102,49,86)(8,114,39,69,27,98,50,82)(9,110,40,65,28,94,51,78)(10,106,41,61,29,105,52,89)(11,117,42,72,30,101,53,85)(12,113,43,68,16,97,54,81)(13,109,44,64,17,93,55,77)(14,120,45,75,18,104,56,88)(15,116,31,71,19,100,57,84), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,16)(13,17)(14,18)(15,19)(31,57)(32,58)(33,59)(34,60)(35,46)(36,47)(37,48)(38,49)(39,50)(40,51)(41,52)(42,53)(43,54)(44,55)(45,56)>;
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,23)(17,22)(18,21)(19,20)(24,30)(25,29)(26,28)(31,32)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(46,54)(47,53)(48,52)(49,51)(55,60)(56,59)(57,58)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(73,75)(76,83)(77,82)(78,81)(79,80)(84,90)(85,89)(86,88)(91,100)(92,99)(93,98)(94,97)(95,96)(101,105)(102,104)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)(118,120), (1,112,32,67,20,96,58,80)(2,108,33,63,21,92,59,76)(3,119,34,74,22,103,60,87)(4,115,35,70,23,99,46,83)(5,111,36,66,24,95,47,79)(6,107,37,62,25,91,48,90)(7,118,38,73,26,102,49,86)(8,114,39,69,27,98,50,82)(9,110,40,65,28,94,51,78)(10,106,41,61,29,105,52,89)(11,117,42,72,30,101,53,85)(12,113,43,68,16,97,54,81)(13,109,44,64,17,93,55,77)(14,120,45,75,18,104,56,88)(15,116,31,71,19,100,57,84), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,16)(13,17)(14,18)(15,19)(31,57)(32,58)(33,59)(34,60)(35,46)(36,47)(37,48)(38,49)(39,50)(40,51)(41,52)(42,53)(43,54)(44,55)(45,56) );
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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,23),(17,22),(18,21),(19,20),(24,30),(25,29),(26,28),(31,32),(33,45),(34,44),(35,43),(36,42),(37,41),(38,40),(46,54),(47,53),(48,52),(49,51),(55,60),(56,59),(57,58),(61,72),(62,71),(63,70),(64,69),(65,68),(66,67),(73,75),(76,83),(77,82),(78,81),(79,80),(84,90),(85,89),(86,88),(91,100),(92,99),(93,98),(94,97),(95,96),(101,105),(102,104),(106,117),(107,116),(108,115),(109,114),(110,113),(111,112),(118,120)], [(1,112,32,67,20,96,58,80),(2,108,33,63,21,92,59,76),(3,119,34,74,22,103,60,87),(4,115,35,70,23,99,46,83),(5,111,36,66,24,95,47,79),(6,107,37,62,25,91,48,90),(7,118,38,73,26,102,49,86),(8,114,39,69,27,98,50,82),(9,110,40,65,28,94,51,78),(10,106,41,61,29,105,52,89),(11,117,42,72,30,101,53,85),(12,113,43,68,16,97,54,81),(13,109,44,64,17,93,55,77),(14,120,45,75,18,104,56,88),(15,116,31,71,19,100,57,84)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,16),(13,17),(14,18),(15,19),(31,57),(32,58),(33,59),(34,60),(35,46),(36,47),(37,48),(38,49),(39,50),(40,51),(41,52),(42,53),(43,54),(44,55),(45,56)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 24C | 24D | 30A | ··· | 30F | 40A | ··· | 40H | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 15 | 15 | 30 | 2 | 1 | 1 | 2 | 15 | 15 | 30 | 2 | 2 | 2 | 4 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 2 | 2 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | ··· | 4 | 12 | ··· | 12 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D5 | D6 | D6 | M4(2) | D10 | D10 | C4×S3 | C4×S3 | C4×D5 | C4×D5 | S3×D5 | S3×M4(2) | D30.C2 | C2×S3×D5 | D30.C2 | D5×M4(2) | D15⋊4M4(2) |
| kernel | D15⋊4M4(2) | D15⋊2C8 | D30.5C4 | C3×C4.Dic5 | C5×C4.Dic3 | C2×C4×D15 | C4×D15 | C2×Dic15 | C22×D15 | C4.Dic5 | C4.Dic3 | C5⋊2C8 | C2×C20 | D15 | C3⋊C8 | C2×C12 | C20 | C2×C10 | C12 | C2×C6 | C2×C4 | C5 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 1 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of D15⋊4M4(2) ►in GL6(𝔽241)
| 0 | 51 | 0 | 0 | 0 | 0 |
| 189 | 189 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 178 | 84 | 0 | 0 | 0 | 0 |
| 79 | 63 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 177 | 0 | 0 |
| 0 | 0 | 177 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 64 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(241))| [0,189,0,0,0,0,51,189,0,0,0,0,0,0,240,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[178,79,0,0,0,0,84,63,0,0,0,0,0,0,240,240,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[64,0,0,0,0,0,0,64,0,0,0,0,0,0,0,177,0,0,0,0,177,0,0,0,0,0,0,0,0,64,0,0,0,0,240,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,1] >;
D15⋊4M4(2) in GAP, Magma, Sage, TeX
D_{15}\rtimes_4M_4(2) % in TeX
G:=Group("D15:4M4(2)"); // GroupNames label
G:=SmallGroup(480,368);
// by ID
G=gap.SmallGroup(480,368);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,64,422,219,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^15=b^2=c^8=d^2=1,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^10*b,b*d=d*b,d*c*d=c^5>;
// generators/relations