Copied to
clipboard

G = M4(2)⋊D15order 480 = 25·3·5

3rd semidirect product of M4(2) and D15 acting via D15/C15=C2

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, C157(C4.D4), (C5×M4(2))⋊9S3, (C3×M4(2))⋊9D5, C60.7C414C2, C22.4(C4×D15), C10.34(D6⋊C4), C4.21(C157D4), C53(C12.46D4), C32(C20.46D4), (C2×C60).55C22, (C22×D15).1C4, C2.9(D303C4), 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

C1C2×C30 — M4(2)⋊D15
C1C5C15C30C60C2×C60C2×D60 — M4(2)⋊D15
C15C30C2×C30 — M4(2)⋊D15
C1C2C2×C4M4(2)

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 [×3], C3, C4 [×2], C22, C22 [×4], C5, S3 [×2], C6, C6, C8 [×2], C2×C4, D4 [×2], C23 [×2], D5 [×2], C10, C10, C12 [×2], D6 [×4], C2×C6, C15, M4(2), M4(2), C2×D4, C20 [×2], D10 [×4], C2×C10, C3⋊C8, C24, D12 [×2], C2×C12, C22×S3 [×2], D15 [×2], C30, C30, C4.D4, C52C8, C40, D20 [×2], C2×C20, C22×D5 [×2], C4.Dic3, C3×M4(2), C2×D12, C60 [×2], D30 [×4], C2×C30, C4.Dic5, C5×M4(2), C2×D20, C12.46D4, C153C8, C120, D60 [×2], C2×C60, C22×D15 [×2], C20.46D4, C60.7C4, C15×M4(2), C2×D60, M4(2)⋊D15
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], 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, C157D4, C20.46D4, D303C4, M4(2)⋊D15

Smallest permutation representation of M4(2)⋊D15
On 120 points
Generators in S120
(1 91 31 84 28 110 46 65)(2 92 32 85 29 111 47 66)(3 93 33 86 30 112 48 67)(4 94 34 87 16 113 49 68)(5 95 35 88 17 114 50 69)(6 96 36 89 18 115 51 70)(7 97 37 90 19 116 52 71)(8 98 38 76 20 117 53 72)(9 99 39 77 21 118 54 73)(10 100 40 78 22 119 55 74)(11 101 41 79 23 120 56 75)(12 102 42 80 24 106 57 61)(13 103 43 81 25 107 58 62)(14 104 44 82 26 108 59 63)(15 105 45 83 27 109 60 64)
(61 80)(62 81)(63 82)(64 83)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 76)(73 77)(74 78)(75 79)(91 110)(92 111)(93 112)(94 113)(95 114)(96 115)(97 116)(98 117)(99 118)(100 119)(101 120)(102 106)(103 107)(104 108)(105 109)
(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 60)(32 59)(33 58)(34 57)(35 56)(36 55)(37 54)(38 53)(39 52)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(61 68)(62 67)(63 66)(64 65)(69 75)(70 74)(71 73)(77 90)(78 89)(79 88)(80 87)(81 86)(82 85)(83 84)(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)

G:=sub<Sym(120)| (1,91,31,84,28,110,46,65)(2,92,32,85,29,111,47,66)(3,93,33,86,30,112,48,67)(4,94,34,87,16,113,49,68)(5,95,35,88,17,114,50,69)(6,96,36,89,18,115,51,70)(7,97,37,90,19,116,52,71)(8,98,38,76,20,117,53,72)(9,99,39,77,21,118,54,73)(10,100,40,78,22,119,55,74)(11,101,41,79,23,120,56,75)(12,102,42,80,24,106,57,61)(13,103,43,81,25,107,58,62)(14,104,44,82,26,108,59,63)(15,105,45,83,27,109,60,64), (61,80)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,76)(73,77)(74,78)(75,79)(91,110)(92,111)(93,112)(94,113)(95,114)(96,115)(97,116)(98,117)(99,118)(100,119)(101,120)(102,106)(103,107)(104,108)(105,109), (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)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(61,68)(62,67)(63,66)(64,65)(69,75)(70,74)(71,73)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,84)(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)>;

G:=Group( (1,91,31,84,28,110,46,65)(2,92,32,85,29,111,47,66)(3,93,33,86,30,112,48,67)(4,94,34,87,16,113,49,68)(5,95,35,88,17,114,50,69)(6,96,36,89,18,115,51,70)(7,97,37,90,19,116,52,71)(8,98,38,76,20,117,53,72)(9,99,39,77,21,118,54,73)(10,100,40,78,22,119,55,74)(11,101,41,79,23,120,56,75)(12,102,42,80,24,106,57,61)(13,103,43,81,25,107,58,62)(14,104,44,82,26,108,59,63)(15,105,45,83,27,109,60,64), (61,80)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,76)(73,77)(74,78)(75,79)(91,110)(92,111)(93,112)(94,113)(95,114)(96,115)(97,116)(98,117)(99,118)(100,119)(101,120)(102,106)(103,107)(104,108)(105,109), (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)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(61,68)(62,67)(63,66)(64,65)(69,75)(70,74)(71,73)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,84)(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) );

G=PermutationGroup([(1,91,31,84,28,110,46,65),(2,92,32,85,29,111,47,66),(3,93,33,86,30,112,48,67),(4,94,34,87,16,113,49,68),(5,95,35,88,17,114,50,69),(6,96,36,89,18,115,51,70),(7,97,37,90,19,116,52,71),(8,98,38,76,20,117,53,72),(9,99,39,77,21,118,54,73),(10,100,40,78,22,119,55,74),(11,101,41,79,23,120,56,75),(12,102,42,80,24,106,57,61),(13,103,43,81,25,107,58,62),(14,104,44,82,26,108,59,63),(15,105,45,83,27,109,60,64)], [(61,80),(62,81),(63,82),(64,83),(65,84),(66,85),(67,86),(68,87),(69,88),(70,89),(71,90),(72,76),(73,77),(74,78),(75,79),(91,110),(92,111),(93,112),(94,113),(95,114),(96,115),(97,116),(98,117),(99,118),(100,119),(101,120),(102,106),(103,107),(104,108),(105,109)], [(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),(31,60),(32,59),(33,58),(34,57),(35,56),(36,55),(37,54),(38,53),(39,52),(40,51),(41,50),(42,49),(43,48),(44,47),(45,46),(61,68),(62,67),(63,66),(64,65),(69,75),(70,74),(71,73),(77,90),(78,89),(79,88),(80,87),(81,86),(82,85),(83,84),(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)])

81 conjugacy classes

class 1 2A2B2C2D 3 4A4B5A5B6A6B8A8B8C8D10A10B10C10D12A12B12C15A15B15C15D20A20B20C20D20E20F24A24B24C24D30A30B30C30D30E30F30G30H40A···40H60A···60H60I60J60K60L120A···120P
order1222234455668888101010101212121515151520202020202024242424303030303030303040···4060···6060606060120···120
size11260602222224446060224422422222222444444222244444···42···244444···4

81 irreducible representations

dim1111122222222222222224444
type++++++++++++++++++
imageC1C2C2C2C4S3D4D5D6D10D12C3⋊D4C4×S3D15D20C5⋊D4C4×D5D30D60C157D4C4×D15C4.D4C12.46D4C20.46D4M4(2)⋊D15
kernelM4(2)⋊D15C60.7C4C15×M4(2)C2×D60C22×D15C5×M4(2)C60C3×M4(2)C2×C20C2×C12C20C20C2×C10M4(2)C12C12C2×C6C2×C4C4C4C22C15C5C3C1
# reps1111412212222444448881248

Matrix representation of M4(2)⋊D15 in GL4(𝔽241) generated by

002401
5124023951
8317810
12418110
,
1000
0100
2341232400
2341230240
,
22511000
1953000
110094110
177131131161
,
639400
12717800
110094110
8414784147
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

׿
×
𝔽