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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, C8⋊3C30, C40⋊7C6, C24⋊7C10, C120⋊15C2, 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

Derived series
C1 — C2 — C15×M4(2)
Chief series
C1 — C2 — C4 — C20 — C60 — C120 — C15×M4(2)
Lower central
C1 — C2 — C15×M4(2)
Upper central
C1 — C60 — C15×M4(2)

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

C1
C2
2C2
C3
C5
C22
C4
C4
C6
2C6
C10
2C10
C15
C8
C2×C4
C8
C12
C12
C2×C6
C20
C20
C2×C10
C30
2C30
M4(2)
C24
C24
C2×C12
C40
C2×C20
C40
C60
C60
C2×C30
C3×M4(2)
C5×M4(2)
C120
C120
C2×C60
C15×M4(2)

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

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

C15×M4(2) is a maximal subgroup of   C60.210D4  M4(2)⋊D15  C4.D60  D60⋊10C4  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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