metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D15⋊2M4(2), C5⋊C8⋊4D6, D15⋊C8⋊6C2, C5⋊3(S3×M4(2)), C15⋊5(C2×M4(2)), C22.F5⋊3S3, C15⋊C8⋊4C22, C22.9(S3×F5), D30.14(C2×C4), C3⋊3(D5⋊M4(2)), Dic3.F5⋊6C2, D30.C2.5C4, (C2×Dic3).6F5, C6.26(C22×F5), C15⋊8M4(2)⋊4C2, C30.26(C22×C4), Dic5.30(C4×S3), Dic3.16(C2×F5), (C10×Dic3).9C4, (C22×D15).6C4, (C2×Dic5).150D6, D30.C2.17C22, (C3×Dic5).36C23, Dic5.38(C22×S3), (C6×Dic5).147C22, C2.26(C2×S3×F5), (C3×C5⋊C8)⋊4C22, C10.26(S3×C2×C4), (C2×C6).8(C2×F5), (C2×C30).21(C2×C4), (C2×C10).21(C4×S3), (C3×C22.F5)⋊4C2, (C2×D30.C2).12C2, (C5×Dic3).15(C2×C4), (C3×Dic5).28(C2×C4), SmallGroup(480,1007)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 692 in 136 conjugacy classes, 48 normal (36 characteristic)
C1, C2, C2 [×4], C3, C4 [×4], C22, C22 [×4], C5, S3 [×3], C6, C6, C8 [×4], C2×C4 [×6], 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⋊C8 [×2], C5⋊C8 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, S3×C8 [×2], C8⋊S3 [×2], C4.Dic3, C3×M4(2), S3×C2×C4, C5×Dic3 [×2], C3×Dic5 [×2], D30 [×2], D30 [×2], C2×C30, D5⋊C8 [×2], C4.F5 [×2], C22.F5, C22.F5, C2×C4×D5, S3×M4(2), C3×C5⋊C8 [×2], C15⋊C8 [×2], D30.C2 [×4], C6×Dic5, C10×Dic3, C22×D15, D5⋊M4(2), D15⋊C8 [×2], Dic3.F5 [×2], C3×C22.F5, C15⋊8M4(2), C2×D30.C2, D15⋊2M4(2)
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D6 [×3], M4(2) [×2], C22×C4, F5, C4×S3 [×2], C22×S3, C2×M4(2), C2×F5 [×3], S3×C2×C4, C22×F5, S3×M4(2), S3×F5, D5⋊M4(2), C2×S3×F5, D15⋊2M4(2)
Generators and relations
G = < a,b,c,d | a15=b2=c8=d2=1, bab=a-1, cac-1=a13, ad=da, cbc-1=a12b, bd=db, dcd=c5 >
►On 120 points
Generators in S120
(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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 30)(10 29)(11 28)(12 27)(13 26)(14 25)(15 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 80)(62 79)(63 78)(64 77)(65 76)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(91 109)(92 108)(93 107)(94 106)(95 120)(96 119)(97 118)(98 117)(99 116)(100 115)(101 114)(102 113)(103 112)(104 111)(105 110)
(1 112 40 61 24 104 46 81)(2 119 44 74 25 96 50 79)(3 111 33 72 26 103 54 77)(4 118 37 70 27 95 58 90)(5 110 41 68 28 102 47 88)(6 117 45 66 29 94 51 86)(7 109 34 64 30 101 55 84)(8 116 38 62 16 93 59 82)(9 108 42 75 17 100 48 80)(10 115 31 73 18 92 52 78)(11 107 35 71 19 99 56 76)(12 114 39 69 20 91 60 89)(13 106 43 67 21 98 49 87)(14 113 32 65 22 105 53 85)(15 120 36 63 23 97 57 83)
(1 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 16)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(31 52)(32 53)(33 54)(34 55)(35 56)(36 57)(37 58)(38 59)(39 60)(40 46)(41 47)(42 48)(43 49)(44 50)(45 51)
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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(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)(61,80)(62,79)(63,78)(64,77)(65,76)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(91,109)(92,108)(93,107)(94,106)(95,120)(96,119)(97,118)(98,117)(99,116)(100,115)(101,114)(102,113)(103,112)(104,111)(105,110), (1,112,40,61,24,104,46,81)(2,119,44,74,25,96,50,79)(3,111,33,72,26,103,54,77)(4,118,37,70,27,95,58,90)(5,110,41,68,28,102,47,88)(6,117,45,66,29,94,51,86)(7,109,34,64,30,101,55,84)(8,116,38,62,16,93,59,82)(9,108,42,75,17,100,48,80)(10,115,31,73,18,92,52,78)(11,107,35,71,19,99,56,76)(12,114,39,69,20,91,60,89)(13,106,43,67,21,98,49,87)(14,113,32,65,22,105,53,85)(15,120,36,63,23,97,57,83), (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,16)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,58)(38,59)(39,60)(40,46)(41,47)(42,48)(43,49)(44,50)(45,51)>;
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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(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)(61,80)(62,79)(63,78)(64,77)(65,76)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(91,109)(92,108)(93,107)(94,106)(95,120)(96,119)(97,118)(98,117)(99,116)(100,115)(101,114)(102,113)(103,112)(104,111)(105,110), (1,112,40,61,24,104,46,81)(2,119,44,74,25,96,50,79)(3,111,33,72,26,103,54,77)(4,118,37,70,27,95,58,90)(5,110,41,68,28,102,47,88)(6,117,45,66,29,94,51,86)(7,109,34,64,30,101,55,84)(8,116,38,62,16,93,59,82)(9,108,42,75,17,100,48,80)(10,115,31,73,18,92,52,78)(11,107,35,71,19,99,56,76)(12,114,39,69,20,91,60,89)(13,106,43,67,21,98,49,87)(14,113,32,65,22,105,53,85)(15,120,36,63,23,97,57,83), (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,16)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,58)(38,59)(39,60)(40,46)(41,47)(42,48)(43,49)(44,50)(45,51) );
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)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,16),(9,30),(10,29),(11,28),(12,27),(13,26),(14,25),(15,24),(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),(61,80),(62,79),(63,78),(64,77),(65,76),(66,90),(67,89),(68,88),(69,87),(70,86),(71,85),(72,84),(73,83),(74,82),(75,81),(91,109),(92,108),(93,107),(94,106),(95,120),(96,119),(97,118),(98,117),(99,116),(100,115),(101,114),(102,113),(103,112),(104,111),(105,110)], [(1,112,40,61,24,104,46,81),(2,119,44,74,25,96,50,79),(3,111,33,72,26,103,54,77),(4,118,37,70,27,95,58,90),(5,110,41,68,28,102,47,88),(6,117,45,66,29,94,51,86),(7,109,34,64,30,101,55,84),(8,116,38,62,16,93,59,82),(9,108,42,75,17,100,48,80),(10,115,31,73,18,92,52,78),(11,107,35,71,19,99,56,76),(12,114,39,69,20,91,60,89),(13,106,43,67,21,98,49,87),(14,113,32,65,22,105,53,85),(15,120,36,63,23,97,57,83)], [(1,24),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,16),(9,17),(10,18),(11,19),(12,20),(13,21),(14,22),(15,23),(31,52),(32,53),(33,54),(34,55),(35,56),(36,57),(37,58),(38,59),(39,60),(40,46),(41,47),(42,48),(43,49),(44,50),(45,51)])
Matrix representation ►G ⊆ GL8(𝔽241)
| 240 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 190 | 51 | 0 | 0 |
| 0 | 0 | 0 | 0 | 190 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 64 | 195 | 0 | 240 |
| 0 | 0 | 0 | 0 | 46 | 195 | 1 | 51 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 51 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 51 | 190 |
| 0 | 0 | 125 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 125 | 0 | 0 | 0 | 0 |
| 200 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 200 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 160 | 115 | 2 | 0 |
| 0 | 0 | 0 | 0 | 160 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 117 | 13 | 81 | 126 |
| 0 | 0 | 0 | 0 | 182 | 13 | 81 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 160 | 115 | 1 | 0 |
| 0 | 0 | 0 | 0 | 160 | 0 | 0 | 1 |
G:=sub<GL(8,GF(241))| [240,240,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,240,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,190,190,64,46,0,0,0,0,51,240,195,195,0,0,0,0,0,0,0,1,0,0,0,0,0,0,240,51],[1,1,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,51,51,0,0,0,0,0,0,1,190],[0,0,200,0,0,0,0,0,0,0,0,200,0,0,0,0,125,0,0,0,0,0,0,0,0,125,0,0,0,0,0,0,0,0,0,0,160,160,117,182,0,0,0,0,115,0,13,13,0,0,0,0,2,0,81,81,0,0,0,0,0,2,126,0],[240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240,0,160,160,0,0,0,0,0,240,115,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 12A | 12B | 12C | 15 | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30A | 30B | 30C |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 |
| size | 1 | 1 | 2 | 15 | 15 | 30 | 2 | 3 | 3 | 5 | 5 | 6 | 10 | 4 | 2 | 4 | 10 | 10 | 10 | 10 | 30 | 30 | 30 | 30 | 4 | 4 | 4 | 10 | 10 | 20 | 8 | 12 | 12 | 12 | 12 | 20 | 20 | 20 | 20 | 8 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D6 | D6 | M4(2) | C4×S3 | C4×S3 | F5 | C2×F5 | C2×F5 | S3×M4(2) | D5⋊M4(2) | S3×F5 | C2×S3×F5 | D15⋊2M4(2) |
| kernel | D15⋊2M4(2) | D15⋊C8 | Dic3.F5 | C3×C22.F5 | C15⋊8M4(2) | C2×D30.C2 | D30.C2 | C10×Dic3 | C22×D15 | C22.F5 | C5⋊C8 | C2×Dic5 | D15 | Dic5 | C2×C10 | C2×Dic3 | Dic3 | C2×C6 | C5 | C3 | C22 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 1 | 4 | 2 | 2 | 1 | 2 | 1 | 2 | 4 | 1 | 1 | 2 |
In GAP, Magma, Sage, TeX
D_{15}\rtimes_2M_{4(2)} % in TeX
G:=Group("D15:2M4(2)"); // GroupNames label
G:=SmallGroup(480,1007);
// by ID
G=gap.SmallGroup(480,1007);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,422,80,1356,9414,2379]);
// 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^13,a*d=d*a,c*b*c^-1=a^12*b,b*d=d*b,d*c*d=c^5>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀