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G = C15×M4(2)  order 240 = 24·3·5

Direct product of C15 and M4(2)

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: C15×M4(2), C4.C60, C83C30, C407C6, C247C10, C12015C2, C22.C60, C20.7C12, C12.4C20, C60.16C4, C60.82C22, C4.6(C2×C30), (C2×C30).5C4, C2.3(C2×C60), (C2×C20).8C6, (C2×C4).2C30, (C2×C6).1C20, (C2×C60).20C2, C30.64(C2×C4), C6.12(C2×C20), (C2×C10).3C12, C20.22(C2×C6), (C2×C12).8C10, C12.22(C2×C10), C10.19(C2×C12), SmallGroup(240,85)

Series: Derived Chief Lower central Upper central

C1C2 — C15×M4(2)
C1C2C4C20C60C120 — C15×M4(2)
C1C2 — C15×M4(2)
C1C60 — C15×M4(2)

Generators and relations for C15×M4(2)
 G = < a,b,c | a15=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

2C2
2C6
2C10
2C30

Smallest permutation representation of C15×M4(2)
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 48 104 31 26 75 112 82)(2 49 105 32 27 61 113 83)(3 50 91 33 28 62 114 84)(4 51 92 34 29 63 115 85)(5 52 93 35 30 64 116 86)(6 53 94 36 16 65 117 87)(7 54 95 37 17 66 118 88)(8 55 96 38 18 67 119 89)(9 56 97 39 19 68 120 90)(10 57 98 40 20 69 106 76)(11 58 99 41 21 70 107 77)(12 59 100 42 22 71 108 78)(13 60 101 43 23 72 109 79)(14 46 102 44 24 73 110 80)(15 47 103 45 25 74 111 81)
(31 82)(32 83)(33 84)(34 85)(35 86)(36 87)(37 88)(38 89)(39 90)(40 76)(41 77)(42 78)(43 79)(44 80)(45 81)(46 73)(47 74)(48 75)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)

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

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

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

C15×M4(2) is a maximal subgroup of   C60.210D4  M4(2)⋊D15  C4.D60  D6010C4  D60.3C4  C8⋊D30  C8.D30

150 conjugacy classes

class 1 2A2B3A3B4A4B4C5A5B5C5D6A6B6C6D8A8B8C8D10A10B10C10D10E10F10G10H12A12B12C12D12E12F15A···15H20A···20H20I20J20K20L24A···24H30A···30H30I···30P40A···40P60A···60P60Q···60X120A···120AF
order12233444555566668888101010101010101012121212121215···1520···202020202024···2430···3030···3040···4060···6060···60120···120
size11211112111111222222111122221111221···11···122222···21···12···22···21···12···22···2

150 irreducible representations

dim111111111111111111112222
type+++
imageC1C2C2C3C4C4C5C6C6C10C10C12C12C15C20C20C30C30C60C60M4(2)C3×M4(2)C5×M4(2)C15×M4(2)
kernelC15×M4(2)C120C2×C60C5×M4(2)C60C2×C30C3×M4(2)C40C2×C20C24C2×C12C20C2×C10M4(2)C12C2×C6C8C2×C4C4C22C15C5C3C1
# reps1212224428444888168161624816

Matrix representation of C15×M4(2) in GL3(𝔽241) generated by

5400
0870
0087
,
100
0178119
011363
,
24000
01176
00240
G:=sub<GL(3,GF(241))| [54,0,0,0,87,0,0,0,87],[1,0,0,0,178,113,0,119,63],[240,0,0,0,1,0,0,176,240] >;

C15×M4(2) in GAP, Magma, Sage, TeX

C_{15}\times M_4(2)
% in TeX

G:=Group("C15xM4(2)");
// GroupNames label

G:=SmallGroup(240,85);
// by ID

G=gap.SmallGroup(240,85);
# by ID

G:=PCGroup([6,-2,-2,-3,-5,-2,-2,360,1465,88]);
// Polycyclic

G:=Group<a,b,c|a^15=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations

Export

Subgroup lattice of C15×M4(2) in TeX

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