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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

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 >

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  D6010C4  D60.3C4  C8⋊D30  C8.D30

150 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 4C 5A 5B 5C 5D 6A 6B 6C 6D 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 10G 10H 12A 12B 12C 12D 12E 12F 15A ··· 15H 20A ··· 20H 20I 20J 20K 20L 24A ··· 24H 30A ··· 30H 30I ··· 30P 40A ··· 40P 60A ··· 60P 60Q ··· 60X 120A ··· 120AF order 1 2 2 3 3 4 4 4 5 5 5 5 6 6 6 6 8 8 8 8 10 10 10 10 10 10 10 10 12 12 12 12 12 12 15 ··· 15 20 ··· 20 20 20 20 20 24 ··· 24 30 ··· 30 30 ··· 30 40 ··· 40 60 ··· 60 60 ··· 60 120 ··· 120 size 1 1 2 1 1 1 1 2 1 1 1 1 1 1 2 2 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 1 ··· 1 1 ··· 1 2 2 2 2 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

150 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + image C1 C2 C2 C3 C4 C4 C5 C6 C6 C10 C10 C12 C12 C15 C20 C20 C30 C30 C60 C60 M4(2) C3×M4(2) C5×M4(2) C15×M4(2) kernel C15×M4(2) C120 C2×C60 C5×M4(2) C60 C2×C30 C3×M4(2) C40 C2×C20 C24 C2×C12 C20 C2×C10 M4(2) C12 C2×C6 C8 C2×C4 C4 C22 C15 C5 C3 C1 # reps 1 2 1 2 2 2 4 4 2 8 4 4 4 8 8 8 16 8 16 16 2 4 8 16

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

 54 0 0 0 87 0 0 0 87
,
 1 0 0 0 178 119 0 113 63
,
 240 0 0 0 1 176 0 0 240
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

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