direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2)×D15, C8⋊6D30, C40⋊18D6, C24⋊21D10, C120⋊25C22, C60.255C23, C5⋊8(S3×M4(2)), C3⋊4(D5×M4(2)), (C8×D15)⋊16C2, C20.61(C4×S3), C60.86(C2×C4), (C4×D15).7C4, C12.29(C4×D5), (C2×C4).45D30, C4.15(C4×D15), C40⋊S3⋊13C2, D30.45(C2×C4), (C2×C20).140D6, (C3×M4(2))⋊5D5, (C5×M4(2))⋊3S3, C15⋊26(C2×M4(2)), C60.7C4⋊17C2, C22.7(C4×D15), C15⋊3C8⋊33C22, (C2×C12).139D10, (C15×M4(2))⋊5C2, (C2×C60).66C22, C4.37(C22×D15), C20.225(C22×S3), C30.166(C22×C4), (C2×Dic15).16C4, Dic15.52(C2×C4), (C4×D15).51C22, (C22×D15).10C4, C12.227(C22×D5), C6.71(C2×C4×D5), (C2×C4×D15).3C2, C2.16(C2×C4×D15), C10.103(S3×C2×C4), (C2×C6).13(C4×D5), (C2×C30).68(C2×C4), (C2×C10).36(C4×S3), SmallGroup(480,871)
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, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 692 in 136 conjugacy classes, 57 normal (41 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C8, C2×C4, C2×C4, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, M4(2), 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), C15⋊3C8, C120, C4×D15, C2×Dic15, C2×C60, C22×D15, D5×M4(2), C8×D15, C40⋊S3, C60.7C4, C15×M4(2), C2×C4×D15, M4(2)×D15
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, M4(2), C22×C4, D10, C4×S3, C22×S3, D15, C2×M4(2), C4×D5, C22×D5, S3×C2×C4, D30, C2×C4×D5, S3×M4(2), C4×D15, C22×D15, D5×M4(2), C2×C4×D15, M4(2)×D15
►On 120 points
(1 91 35 76 16 106 47 62)(2 92 36 77 17 107 48 63)(3 93 37 78 18 108 49 64)(4 94 38 79 19 109 50 65)(5 95 39 80 20 110 51 66)(6 96 40 81 21 111 52 67)(7 97 41 82 22 112 53 68)(8 98 42 83 23 113 54 69)(9 99 43 84 24 114 55 70)(10 100 44 85 25 115 56 71)(11 101 45 86 26 116 57 72)(12 102 31 87 27 117 58 73)(13 103 32 88 28 118 59 74)(14 104 33 89 29 119 60 75)(15 105 34 90 30 120 46 61)
(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)(91 106)(92 107)(93 108)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)(105 120)
(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 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(31 38)(32 37)(33 36)(34 35)(39 45)(40 44)(41 43)(46 47)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(61 62)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(76 90)(77 89)(78 88)(79 87)(80 86)(81 85)(82 84)(91 105)(92 104)(93 103)(94 102)(95 101)(96 100)(97 99)(106 120)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)
G:=sub<Sym(120)| (1,91,35,76,16,106,47,62)(2,92,36,77,17,107,48,63)(3,93,37,78,18,108,49,64)(4,94,38,79,19,109,50,65)(5,95,39,80,20,110,51,66)(6,96,40,81,21,111,52,67)(7,97,41,82,22,112,53,68)(8,98,42,83,23,113,54,69)(9,99,43,84,24,114,55,70)(10,100,44,85,25,115,56,71)(11,101,45,86,26,116,57,72)(12,102,31,87,27,117,58,73)(13,103,32,88,28,118,59,74)(14,104,33,89,29,119,60,75)(15,105,34,90,30,120,46,61), (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)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120), (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)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(31,38)(32,37)(33,36)(34,35)(39,45)(40,44)(41,43)(46,47)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(61,62)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(91,105)(92,104)(93,103)(94,102)(95,101)(96,100)(97,99)(106,120)(107,119)(108,118)(109,117)(110,116)(111,115)(112,114)>;
G:=Group( (1,91,35,76,16,106,47,62)(2,92,36,77,17,107,48,63)(3,93,37,78,18,108,49,64)(4,94,38,79,19,109,50,65)(5,95,39,80,20,110,51,66)(6,96,40,81,21,111,52,67)(7,97,41,82,22,112,53,68)(8,98,42,83,23,113,54,69)(9,99,43,84,24,114,55,70)(10,100,44,85,25,115,56,71)(11,101,45,86,26,116,57,72)(12,102,31,87,27,117,58,73)(13,103,32,88,28,118,59,74)(14,104,33,89,29,119,60,75)(15,105,34,90,30,120,46,61), (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)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120), (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)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(31,38)(32,37)(33,36)(34,35)(39,45)(40,44)(41,43)(46,47)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(61,62)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(91,105)(92,104)(93,103)(94,102)(95,101)(96,100)(97,99)(106,120)(107,119)(108,118)(109,117)(110,116)(111,115)(112,114) );
G=PermutationGroup([[(1,91,35,76,16,106,47,62),(2,92,36,77,17,107,48,63),(3,93,37,78,18,108,49,64),(4,94,38,79,19,109,50,65),(5,95,39,80,20,110,51,66),(6,96,40,81,21,111,52,67),(7,97,41,82,22,112,53,68),(8,98,42,83,23,113,54,69),(9,99,43,84,24,114,55,70),(10,100,44,85,25,115,56,71),(11,101,45,86,26,116,57,72),(12,102,31,87,27,117,58,73),(13,103,32,88,28,118,59,74),(14,104,33,89,29,119,60,75),(15,105,34,90,30,120,46,61)], [(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),(91,106),(92,107),(93,108),(94,109),(95,110),(96,111),(97,112),(98,113),(99,114),(100,115),(101,116),(102,117),(103,118),(104,119),(105,120)], [(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),(16,30),(17,29),(18,28),(19,27),(20,26),(21,25),(22,24),(31,38),(32,37),(33,36),(34,35),(39,45),(40,44),(41,43),(46,47),(48,60),(49,59),(50,58),(51,57),(52,56),(53,55),(61,62),(63,75),(64,74),(65,73),(66,72),(67,71),(68,70),(76,90),(77,89),(78,88),(79,87),(80,86),(81,85),(82,84),(91,105),(92,104),(93,103),(94,102),(95,101),(96,100),(97,99),(106,120),(107,119),(108,118),(109,117),(110,116),(111,115),(112,114)]])
90 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 | 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 | 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 | 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 | 15 | 15 | 30 | 2 | 1 | 1 | 2 | 15 | 15 | 30 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 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 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | D15 | C4×D5 | C4×D5 | D30 | D30 | C4×D15 | C4×D15 | S3×M4(2) | D5×M4(2) | M4(2)×D15 |
| kernel | M4(2)×D15 | C8×D15 | C40⋊S3 | C60.7C4 | C15×M4(2) | C2×C4×D15 | C4×D15 | C2×Dic15 | C22×D15 | C5×M4(2) | C3×M4(2) | C40 | C2×C20 | D15 | C24 | C2×C12 | C20 | C2×C10 | M4(2) | C12 | C2×C6 | C8 | C2×C4 | C4 | C22 | C5 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 1 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 4 | 8 | 8 | 2 | 4 | 8 |
Matrix representation of M4(2)×D15 ►in GL4(𝔽241) generated by
| 177 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 |
| 0 | 0 | 240 | 239 |
| 0 | 0 | 153 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 240 | 240 |
| 68 | 80 | 0 | 0 |
| 98 | 211 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 240 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(241))| [177,0,0,0,0,177,0,0,0,0,240,153,0,0,239,1],[1,0,0,0,0,1,0,0,0,0,1,240,0,0,0,240],[68,98,0,0,80,211,0,0,0,0,1,0,0,0,0,1],[240,0,0,0,1,1,0,0,0,0,1,0,0,0,0,1] >;
M4(2)×D15 in GAP, Magma, Sage, TeX
M_4(2)\times D_{15} % in TeX
G:=Group("M4(2)xD15"); // GroupNames label
G:=SmallGroup(480,871);
// by ID
G=gap.SmallGroup(480,871);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,58,80,2693,18822]);
// 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,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations