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## G = C3×C4.Dic5order 240 = 24·3·5

### Direct product of C3 and C4.Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×C4.Dic5
 Chief series C1 — C5 — C10 — C20 — C60 — C3×C5⋊2C8 — C3×C4.Dic5
 Lower central C5 — C10 — C3×C4.Dic5
 Upper central C1 — C12 — C2×C12

Generators and relations for C3×C4.Dic5
G = < a,b,c,d | a3=b4=1, c10=b2, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c9 >

Smallest permutation representation of C3×C4.Dic5
On 120 points
Generators in S120
(1 56 31)(2 57 32)(3 58 33)(4 59 34)(5 60 35)(6 41 36)(7 42 37)(8 43 38)(9 44 39)(10 45 40)(11 46 21)(12 47 22)(13 48 23)(14 49 24)(15 50 25)(16 51 26)(17 52 27)(18 53 28)(19 54 29)(20 55 30)(61 118 87)(62 119 88)(63 120 89)(64 101 90)(65 102 91)(66 103 92)(67 104 93)(68 105 94)(69 106 95)(70 107 96)(71 108 97)(72 109 98)(73 110 99)(74 111 100)(75 112 81)(76 113 82)(77 114 83)(78 115 84)(79 116 85)(80 117 86)
(1 6 11 16)(2 7 12 17)(3 8 13 18)(4 9 14 19)(5 10 15 20)(21 26 31 36)(22 27 32 37)(23 28 33 38)(24 29 34 39)(25 30 35 40)(41 46 51 56)(42 47 52 57)(43 48 53 58)(44 49 54 59)(45 50 55 60)(61 76 71 66)(62 77 72 67)(63 78 73 68)(64 79 74 69)(65 80 75 70)(81 96 91 86)(82 97 92 87)(83 98 93 88)(84 99 94 89)(85 100 95 90)(101 116 111 106)(102 117 112 107)(103 118 113 108)(104 119 114 109)(105 120 115 110)
(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 72 6 77 11 62 16 67)(2 61 7 66 12 71 17 76)(3 70 8 75 13 80 18 65)(4 79 9 64 14 69 19 74)(5 68 10 73 15 78 20 63)(21 88 26 93 31 98 36 83)(22 97 27 82 32 87 37 92)(23 86 28 91 33 96 38 81)(24 95 29 100 34 85 39 90)(25 84 30 89 35 94 40 99)(41 114 46 119 51 104 56 109)(42 103 47 108 52 113 57 118)(43 112 48 117 53 102 58 107)(44 101 49 106 54 111 59 116)(45 110 50 115 55 120 60 105)

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

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

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

C3×C4.Dic5 is a maximal subgroup of
C20.5D12  C60.29D4  C60.54D4  C60.31D4  C60.98D4  D6013C4  C60.D4  C12.59D20  D12.Dic5  D60.4C4  D154M4(2)  D6036C22  C60.38D4  C20.D12  D12.33D10  C3×D5×M4(2)

78 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 4C 5A 5B 6A 6B 6C 6D 8A 8B 8C 8D 10A ··· 10F 12A 12B 12C 12D 12E 12F 15A 15B 15C 15D 20A ··· 20H 24A ··· 24H 30A ··· 30L 60A ··· 60P order 1 2 2 3 3 4 4 4 5 5 6 6 6 6 8 8 8 8 10 ··· 10 12 12 12 12 12 12 15 15 15 15 20 ··· 20 24 ··· 24 30 ··· 30 60 ··· 60 size 1 1 2 1 1 1 1 2 2 2 1 1 2 2 10 10 10 10 2 ··· 2 1 1 1 1 2 2 2 2 2 2 2 ··· 2 10 ··· 10 2 ··· 2 2 ··· 2

78 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 D5 M4(2) Dic5 D10 Dic5 C3×D5 C3×M4(2) C3×Dic5 C6×D5 C3×Dic5 C4.Dic5 C3×C4.Dic5 kernel C3×C4.Dic5 C3×C5⋊2C8 C2×C60 C4.Dic5 C60 C2×C30 C5⋊2C8 C2×C20 C20 C2×C10 C2×C12 C15 C12 C12 C2×C6 C2×C4 C5 C4 C4 C22 C3 C1 # reps 1 2 1 2 2 2 4 2 4 4 2 2 2 2 2 4 4 4 4 4 8 16

Matrix representation of C3×C4.Dic5 in GL3(𝔽241) generated by

 225 0 0 0 1 0 0 0 1
,
 240 0 0 0 64 0 0 0 177
,
 1 0 0 0 6 0 0 0 40
,
 1 0 0 0 0 1 0 64 0
G:=sub<GL(3,GF(241))| [225,0,0,0,1,0,0,0,1],[240,0,0,0,64,0,0,0,177],[1,0,0,0,6,0,0,0,40],[1,0,0,0,0,64,0,1,0] >;

C3×C4.Dic5 in GAP, Magma, Sage, TeX

C_3\times C_4.{\rm Dic}_5
% in TeX

G:=Group("C3xC4.Dic5");
// GroupNames label

G:=SmallGroup(240,39);
// by ID

G=gap.SmallGroup(240,39);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-5,72,313,69,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=1,c^10=b^2,d^2=c^5,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^9>;
// generators/relations

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