direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C4.Dic5, C20.4C12, C60.10C4, C12.59D10, C15⋊14M4(2), C12.4Dic5, C60.72C22, C5⋊2C8⋊5C6, C4.(C3×Dic5), (C2×C30).9C4, (C2×C20).5C6, (C2×C12).8D5, C4.15(C6×D5), C5⋊4(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×C5⋊2C8)⋊12C2, (C2×C4).2(C3×D5), SmallGroup(240,39)
Series: Derived ►Chief ►Lower central ►Upper central
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 51 26)(2 52 27)(3 53 28)(4 54 29)(5 55 30)(6 56 31)(7 57 32)(8 58 33)(9 59 34)(10 60 35)(11 41 36)(12 42 37)(13 43 38)(14 44 39)(15 45 40)(16 46 21)(17 47 22)(18 48 23)(19 49 24)(20 50 25)(61 117 96)(62 118 97)(63 119 98)(64 120 99)(65 101 100)(66 102 81)(67 103 82)(68 104 83)(69 105 84)(70 106 85)(71 107 86)(72 108 87)(73 109 88)(74 110 89)(75 111 90)(76 112 91)(77 113 92)(78 114 93)(79 115 94)(80 116 95)
(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 78 6 63 11 68 16 73)(2 67 7 72 12 77 17 62)(3 76 8 61 13 66 18 71)(4 65 9 70 14 75 19 80)(5 74 10 79 15 64 20 69)(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)(41 104 46 109 51 114 56 119)(42 113 47 118 52 103 57 108)(43 102 48 107 53 112 58 117)(44 111 49 116 54 101 59 106)(45 120 50 105 55 110 60 115)
G:=sub<Sym(120)| (1,51,26)(2,52,27)(3,53,28)(4,54,29)(5,55,30)(6,56,31)(7,57,32)(8,58,33)(9,59,34)(10,60,35)(11,41,36)(12,42,37)(13,43,38)(14,44,39)(15,45,40)(16,46,21)(17,47,22)(18,48,23)(19,49,24)(20,50,25)(61,117,96)(62,118,97)(63,119,98)(64,120,99)(65,101,100)(66,102,81)(67,103,82)(68,104,83)(69,105,84)(70,106,85)(71,107,86)(72,108,87)(73,109,88)(74,110,89)(75,111,90)(76,112,91)(77,113,92)(78,114,93)(79,115,94)(80,116,95), (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), (1,78,6,63,11,68,16,73)(2,67,7,72,12,77,17,62)(3,76,8,61,13,66,18,71)(4,65,9,70,14,75,19,80)(5,74,10,79,15,64,20,69)(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)(41,104,46,109,51,114,56,119)(42,113,47,118,52,103,57,108)(43,102,48,107,53,112,58,117)(44,111,49,116,54,101,59,106)(45,120,50,105,55,110,60,115)>;
G:=Group( (1,51,26)(2,52,27)(3,53,28)(4,54,29)(5,55,30)(6,56,31)(7,57,32)(8,58,33)(9,59,34)(10,60,35)(11,41,36)(12,42,37)(13,43,38)(14,44,39)(15,45,40)(16,46,21)(17,47,22)(18,48,23)(19,49,24)(20,50,25)(61,117,96)(62,118,97)(63,119,98)(64,120,99)(65,101,100)(66,102,81)(67,103,82)(68,104,83)(69,105,84)(70,106,85)(71,107,86)(72,108,87)(73,109,88)(74,110,89)(75,111,90)(76,112,91)(77,113,92)(78,114,93)(79,115,94)(80,116,95), (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), (1,78,6,63,11,68,16,73)(2,67,7,72,12,77,17,62)(3,76,8,61,13,66,18,71)(4,65,9,70,14,75,19,80)(5,74,10,79,15,64,20,69)(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)(41,104,46,109,51,114,56,119)(42,113,47,118,52,103,57,108)(43,102,48,107,53,112,58,117)(44,111,49,116,54,101,59,106)(45,120,50,105,55,110,60,115) );
G=PermutationGroup([[(1,51,26),(2,52,27),(3,53,28),(4,54,29),(5,55,30),(6,56,31),(7,57,32),(8,58,33),(9,59,34),(10,60,35),(11,41,36),(12,42,37),(13,43,38),(14,44,39),(15,45,40),(16,46,21),(17,47,22),(18,48,23),(19,49,24),(20,50,25),(61,117,96),(62,118,97),(63,119,98),(64,120,99),(65,101,100),(66,102,81),(67,103,82),(68,104,83),(69,105,84),(70,106,85),(71,107,86),(72,108,87),(73,109,88),(74,110,89),(75,111,90),(76,112,91),(77,113,92),(78,114,93),(79,115,94),(80,116,95)], [(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)], [(1,78,6,63,11,68,16,73),(2,67,7,72,12,77,17,62),(3,76,8,61,13,66,18,71),(4,65,9,70,14,75,19,80),(5,74,10,79,15,64,20,69),(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),(41,104,46,109,51,114,56,119),(42,113,47,118,52,103,57,108),(43,102,48,107,53,112,58,117),(44,111,49,116,54,101,59,106),(45,120,50,105,55,110,60,115)]])
C3×C4.Dic5 is a maximal subgroup of
C20.5D12 C60.29D4 C60.54D4 C60.31D4 C60.98D4 D60⋊13C4 C60.D4 C12.59D20 D12.Dic5 D60.4C4 D15⋊4M4(2) D60⋊36C22 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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