direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C4.F5, C60.4C4, C12.4F5, C20.1C12, C15⋊4M4(2), D10.3C12, C5⋊C8⋊1C6, C4.(C3×F5), C2.4(C6×F5), (C6×D5).7C4, (C4×D5).3C6, C5⋊1(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
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 >
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
D12⋊4F5 D60⋊2C4 D10.Dic6 D10.2Dic6 D12.F5 D15⋊M4(2) Dic6.F5
42 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 5 | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 10 | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | 20B | 24A | ··· | 24H | 30A | 30B | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 24 | ··· | 24 | 30 | 30 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 10 | 1 | 1 | 2 | 5 | 5 | 4 | 1 | 1 | 10 | 10 | 10 | 10 | 10 | 10 | 4 | 2 | 2 | 5 | 5 | 5 | 5 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | M4(2) | C3×M4(2) | F5 | C2×F5 | C3×F5 | C4.F5 | C6×F5 | C3×C4.F5 |
| kernel | C3×C4.F5 | C3×C5⋊C8 | D5×C12 | C4.F5 | C60 | C6×D5 | C5⋊C8 | C4×D5 | C20 | D10 | C15 | C5 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 2 | 4 | 1 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of C3×C4.F5 ►in GL4(𝔽7) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 5 | 4 | 4 | 4 |
| 5 | 1 | 0 | 3 |
| 2 | 6 | 4 | 2 |
| 4 | 2 | 5 | 4 |
| 1 | 0 | 2 | 0 |
| 0 | 4 | 6 | 0 |
| 6 | 2 | 0 | 1 |
| 2 | 5 | 5 | 1 |
| 5 | 3 | 3 | 2 |
| 6 | 6 | 5 | 1 |
| 4 | 2 | 2 | 4 |
| 0 | 5 | 5 | 1 |
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
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