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 [×4], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, S3 [×3], C6, C6, C8 [×4], C2×C4, C2×C4 [×5], C23, D5 [×3], C10, C10, Dic3 [×2], C12 [×2], D6 [×4], C2×C6, C15, C2×C8 [×2], M4(2) [×4], C22×C4, Dic5 [×2], C20 [×2], D10 [×4], C2×C10, C3⋊C8 [×2], C24 [×2], C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, D15 [×2], D15, C30, C30, C2×M4(2), C5⋊2C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, S3×C8 [×2], C8⋊S3 [×2], C4.Dic3, C3×M4(2), S3×C2×C4, Dic15 [×2], C60 [×2], D30 [×2], D30 [×2], C2×C30, C8×D5 [×2], C8⋊D5 [×2], C4.Dic5, C5×M4(2), C2×C4×D5, S3×M4(2), C5×C3⋊C8 [×2], C3×C5⋊2C8 [×2], C4×D15 [×4], C2×Dic15, C2×C60, C22×D15, D5×M4(2), D15⋊2C8 [×2], D30.5C4 [×2], C3×C4.Dic5, C5×C4.Dic3, C2×C4×D15, D15⋊4M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, D6 [×3], M4(2) [×2], C22×C4, D10 [×3], C4×S3 [×2], C22×S3, C2×M4(2), C4×D5 [×2], C22×D5, S3×C2×C4, S3×D5, C2×C4×D5, S3×M4(2), D30.C2 [×2], 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 24)(17 23)(18 22)(19 21)(25 30)(26 29)(27 28)(31 39)(32 38)(33 37)(34 36)(40 45)(41 44)(42 43)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(61 74)(62 73)(63 72)(64 71)(65 70)(66 69)(67 68)(76 77)(78 90)(79 89)(80 88)(81 87)(82 86)(83 85)(91 101)(92 100)(93 99)(94 98)(95 97)(102 105)(103 104)(106 108)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)
(1 115 43 68 28 104 46 77)(2 111 44 64 29 100 47 88)(3 107 45 75 30 96 48 84)(4 118 31 71 16 92 49 80)(5 114 32 67 17 103 50 76)(6 110 33 63 18 99 51 87)(7 106 34 74 19 95 52 83)(8 117 35 70 20 91 53 79)(9 113 36 66 21 102 54 90)(10 109 37 62 22 98 55 86)(11 120 38 73 23 94 56 82)(12 116 39 69 24 105 57 78)(13 112 40 65 25 101 58 89)(14 108 41 61 26 97 59 85)(15 119 42 72 27 93 60 81)
(1 28)(2 29)(3 30)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 46)(44 47)(45 48)
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)(16,24)(17,23)(18,22)(19,21)(25,30)(26,29)(27,28)(31,39)(32,38)(33,37)(34,36)(40,45)(41,44)(42,43)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)(76,77)(78,90)(79,89)(80,88)(81,87)(82,86)(83,85)(91,101)(92,100)(93,99)(94,98)(95,97)(102,105)(103,104)(106,108)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115), (1,115,43,68,28,104,46,77)(2,111,44,64,29,100,47,88)(3,107,45,75,30,96,48,84)(4,118,31,71,16,92,49,80)(5,114,32,67,17,103,50,76)(6,110,33,63,18,99,51,87)(7,106,34,74,19,95,52,83)(8,117,35,70,20,91,53,79)(9,113,36,66,21,102,54,90)(10,109,37,62,22,98,55,86)(11,120,38,73,23,94,56,82)(12,116,39,69,24,105,57,78)(13,112,40,65,25,101,58,89)(14,108,41,61,26,97,59,85)(15,119,42,72,27,93,60,81), (1,28)(2,29)(3,30)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,46)(44,47)(45,48)>;
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)(16,24)(17,23)(18,22)(19,21)(25,30)(26,29)(27,28)(31,39)(32,38)(33,37)(34,36)(40,45)(41,44)(42,43)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)(76,77)(78,90)(79,89)(80,88)(81,87)(82,86)(83,85)(91,101)(92,100)(93,99)(94,98)(95,97)(102,105)(103,104)(106,108)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115), (1,115,43,68,28,104,46,77)(2,111,44,64,29,100,47,88)(3,107,45,75,30,96,48,84)(4,118,31,71,16,92,49,80)(5,114,32,67,17,103,50,76)(6,110,33,63,18,99,51,87)(7,106,34,74,19,95,52,83)(8,117,35,70,20,91,53,79)(9,113,36,66,21,102,54,90)(10,109,37,62,22,98,55,86)(11,120,38,73,23,94,56,82)(12,116,39,69,24,105,57,78)(13,112,40,65,25,101,58,89)(14,108,41,61,26,97,59,85)(15,119,42,72,27,93,60,81), (1,28)(2,29)(3,30)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,46)(44,47)(45,48) );
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),(16,24),(17,23),(18,22),(19,21),(25,30),(26,29),(27,28),(31,39),(32,38),(33,37),(34,36),(40,45),(41,44),(42,43),(46,60),(47,59),(48,58),(49,57),(50,56),(51,55),(52,54),(61,74),(62,73),(63,72),(64,71),(65,70),(66,69),(67,68),(76,77),(78,90),(79,89),(80,88),(81,87),(82,86),(83,85),(91,101),(92,100),(93,99),(94,98),(95,97),(102,105),(103,104),(106,108),(109,120),(110,119),(111,118),(112,117),(113,116),(114,115)], [(1,115,43,68,28,104,46,77),(2,111,44,64,29,100,47,88),(3,107,45,75,30,96,48,84),(4,118,31,71,16,92,49,80),(5,114,32,67,17,103,50,76),(6,110,33,63,18,99,51,87),(7,106,34,74,19,95,52,83),(8,117,35,70,20,91,53,79),(9,113,36,66,21,102,54,90),(10,109,37,62,22,98,55,86),(11,120,38,73,23,94,56,82),(12,116,39,69,24,105,57,78),(13,112,40,65,25,101,58,89),(14,108,41,61,26,97,59,85),(15,119,42,72,27,93,60,81)], [(1,28),(2,29),(3,30),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(13,25),(14,26),(15,27),(31,49),(32,50),(33,51),(34,52),(35,53),(36,54),(37,55),(38,56),(39,57),(40,58),(41,59),(42,60),(43,46),(44,47),(45,48)])
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