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G = C3×C4.F5order 240 = 24·3·5

Direct product of C3 and C4.F5

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

Aliases: C3×C4.F5, C60.4C4, C12.4F5, C20.1C12, C154M4(2), D10.3C12, C5⋊C81C6, C4.(C3×F5), C2.4(C6×F5), (C6×D5).7C4, (C4×D5).3C6, C51(C3×M4(2)), C6.16(C2×F5), C30.16(C2×C4), C10.2(C2×C12), (D5×C12).10C2, Dic5.6(C2×C6), (C3×Dic5).26C22, (C3×C5⋊C8)⋊5C2, SmallGroup(240,112)

Series: Derived Chief Lower central Upper central

C1C10 — C3×C4.F5
C1C5C10Dic5C3×Dic5C3×C5⋊C8 — C3×C4.F5
C5C10 — C3×C4.F5
C1C6C12

Generators and relations for C3×C4.F5
 G = < a,b,c,d | a3=b4=c5=1, d4=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >

10C2
5C4
5C22
10C6
2D5
5C8
5C2×C4
5C8
5C2×C6
5C12
2C3×D5
5M4(2)
5C24
5C24
5C2×C12
5C3×M4(2)

Smallest permutation representation of C3×C4.F5
On 120 points
Generators in S120
(1 100 60)(2 101 61)(3 102 62)(4 103 63)(5 104 64)(6 97 57)(7 98 58)(8 99 59)(9 86 46)(10 87 47)(11 88 48)(12 81 41)(13 82 42)(14 83 43)(15 84 44)(16 85 45)(17 95 120)(18 96 113)(19 89 114)(20 90 115)(21 91 116)(22 92 117)(23 93 118)(24 94 119)(25 110 66)(26 111 67)(27 112 68)(28 105 69)(29 106 70)(30 107 71)(31 108 72)(32 109 65)(33 56 74)(34 49 75)(35 50 76)(36 51 77)(37 52 78)(38 53 79)(39 54 80)(40 55 73)
(1 7 5 3)(2 4 6 8)(9 90 13 94)(10 95 14 91)(11 92 15 96)(12 89 16 93)(17 43 21 47)(18 48 22 44)(19 45 23 41)(20 42 24 46)(25 33 29 37)(26 38 30 34)(27 35 31 39)(28 40 32 36)(49 111 53 107)(50 108 54 112)(51 105 55 109)(52 110 56 106)(57 59 61 63)(58 64 62 60)(65 77 69 73)(66 74 70 78)(67 79 71 75)(68 76 72 80)(81 114 85 118)(82 119 86 115)(83 116 87 120)(84 113 88 117)(97 99 101 103)(98 104 102 100)
(1 49 46 22 105)(2 23 50 106 47)(3 107 24 48 51)(4 41 108 52 17)(5 53 42 18 109)(6 19 54 110 43)(7 111 20 44 55)(8 45 112 56 21)(9 92 69 100 75)(10 101 93 76 70)(11 77 102 71 94)(12 72 78 95 103)(13 96 65 104 79)(14 97 89 80 66)(15 73 98 67 90)(16 68 74 91 99)(25 83 57 114 39)(26 115 84 40 58)(27 33 116 59 85)(28 60 34 86 117)(29 87 61 118 35)(30 119 88 36 62)(31 37 120 63 81)(32 64 38 82 113)
(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)

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

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

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

C3×C4.F5 is a maximal subgroup of   D124F5  D602C4  D10.Dic6  D10.2Dic6  D12.F5  D15⋊M4(2)  Dic6.F5

42 conjugacy classes

class 1 2A2B3A3B4A4B4C 5 6A6B6C6D8A8B8C8D 10 12A12B12C12D12E12F15A15B20A20B24A···24H30A30B60A60B60C60D
order12233444566668888101212121212121515202024···24303060606060
size1110112554111010101010104225555444410···10444444

42 irreducible representations

dim111111111122444444
type+++++
imageC1C2C2C3C4C4C6C6C12C12M4(2)C3×M4(2)F5C2×F5C3×F5C4.F5C6×F5C3×C4.F5
kernelC3×C4.F5C3×C5⋊C8D5×C12C4.F5C60C6×D5C5⋊C8C4×D5C20D10C15C5C12C6C4C3C2C1
# reps121222424424112224

Matrix representation of C3×C4.F5 in GL4(𝔽7) generated by

2000
0200
0020
0002
,
5444
5103
2642
4254
,
1020
0460
6201
2551
,
5332
6651
4224
0551
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[5,5,2,4,4,1,6,2,4,0,4,5,4,3,2,4],[1,0,6,2,0,4,2,5,2,6,0,5,0,0,1,1],[5,6,4,0,3,6,2,5,3,5,2,5,2,1,4,1] >;

C3×C4.F5 in GAP, Magma, Sage, TeX

C_3\times C_4.F_5
% in TeX

G:=Group("C3xC4.F5");
// GroupNames label

G:=SmallGroup(240,112);
// by ID

G=gap.SmallGroup(240,112);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-5,72,313,151,69,3461,599]);
// Polycyclic

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

Export

Subgroup lattice of C3×C4.F5 in TeX

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