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G = M4(2)×D15order 480 = 25·3·5

Direct product of M4(2) and D15

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M4(2)×D15, C86D30, C4018D6, C2421D10, C12025C22, C60.255C23, C58(S3×M4(2)), C34(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⋊S313C2, D30.45(C2×C4), (C2×C20).140D6, (C3×M4(2))⋊5D5, (C5×M4(2))⋊3S3, C1526(C2×M4(2)), C60.7C417C2, C22.7(C4×D15), C153C833C22, (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

C1C30 — M4(2)×D15
C1C5C15C30C60C4×D15C2×C4×D15 — M4(2)×D15
C15C30 — M4(2)×D15
C1C4M4(2)

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 [×4], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, S3 [×3], C6, C6, C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, D5 [×3], C10, C10, Dic3 [×2], C12 [×2], D6 [×4], C2×C6, C15, C2×C8 [×2], M4(2), M4(2) [×3], 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), C52C8 [×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), C153C8 [×2], C120 [×2], C4×D15 [×4], C2×Dic15, C2×C60, C22×D15, D5×M4(2), C8×D15 [×2], C40⋊S3 [×2], C60.7C4, C15×M4(2), C2×C4×D15, M4(2)×D15
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, D15, C2×M4(2), C4×D5 [×2], C22×D5, S3×C2×C4, D30 [×3], C2×C4×D5, S3×M4(2), C4×D15 [×2], C22×D15, D5×M4(2), C2×C4×D15, M4(2)×D15

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

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

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

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

90 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F5A5B6A6B8A8B8C8D8E8F8G8H10A10B10C10D12A12B12C15A15B15C15D20A20B20C20D20E20F24A24B24C24D30A30B30C30D30E30F30G30H40A···40H60A···60H60I60J60K60L120A···120P
order1222223444444556688888888101010101212121515151520202020202024242424303030303030303040···4060···6060606060120···120
size11215153021121515302224222230303030224422422222222444444222244444···42···244444···4

90 irreducible representations

dim1111111112222222222222222444
type+++++++++++++++
imageC1C2C2C2C2C2C4C4C4S3D5D6D6M4(2)D10D10C4×S3C4×S3D15C4×D5C4×D5D30D30C4×D15C4×D15S3×M4(2)D5×M4(2)M4(2)×D15
kernelM4(2)×D15C8×D15C40⋊S3C60.7C4C15×M4(2)C2×C4×D15C4×D15C2×Dic15C22×D15C5×M4(2)C3×M4(2)C40C2×C20D15C24C2×C12C20C2×C10M4(2)C12C2×C6C8C2×C4C4C22C5C3C1
# reps1221114221221442224448488248

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

177000
017700
00240239
001531
,
1000
0100
0010
00240240
,
688000
9821100
0010
0001
,
240100
0100
0010
0001
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

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