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G = C3×C4.Dic5order 240 = 24·3·5

Direct product of C3 and C4.Dic5

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

Aliases: C3×C4.Dic5, C20.4C12, C60.10C4, C12.59D10, C1514M4(2), C12.4Dic5, C60.72C22, C52C85C6, C4.(C3×Dic5), (C2×C30).9C4, (C2×C20).5C6, (C2×C12).8D5, C4.15(C6×D5), C54(C3×M4(2)), (C2×C60).12C2, (C2×C10).5C12, C20.16(C2×C6), C30.55(C2×C4), (C2×C6).1Dic5, C2.3(C6×Dic5), C10.13(C2×C12), C22.(C3×Dic5), C6.12(C2×Dic5), (C3×C52C8)⋊12C2, (C2×C4).2(C3×D5), SmallGroup(240,39)

Series: Derived Chief Lower central Upper central

C1C10 — C3×C4.Dic5
C1C5C10C20C60C3×C52C8 — C3×C4.Dic5
C5C10 — C3×C4.Dic5
C1C12C2×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 >

2C2
2C6
2C10
5C8
5C8
2C30
5M4(2)
5C24
5C24
5C3×M4(2)

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 2A2B3A3B4A4B4C5A5B6A6B6C6D8A8B8C8D10A···10F12A12B12C12D12E12F15A15B15C15D20A···20H24A···24H30A···30L60A···60P
order12233444556666888810···101212121212121515151520···2024···2430···3060···60
size11211112221122101010102···211112222222···210···102···22···2

78 irreducible representations

dim1111111111222222222222
type++++-+-
imageC1C2C2C3C4C4C6C6C12C12D5M4(2)Dic5D10Dic5C3×D5C3×M4(2)C3×Dic5C6×D5C3×Dic5C4.Dic5C3×C4.Dic5
kernelC3×C4.Dic5C3×C52C8C2×C60C4.Dic5C60C2×C30C52C8C2×C20C20C2×C10C2×C12C15C12C12C2×C6C2×C4C5C4C4C22C3C1
# reps12122242442222244444816

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

22500
010
001
,
24000
0640
00177
,
100
060
0040
,
100
001
0640
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

Export

Subgroup lattice of C3×C4.Dic5 in TeX

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