direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6×C4⋊F5, C4⋊2(C6×F5), (C2×C60)⋊8C4, C60⋊9(C2×C4), (C2×C12)⋊8F5, C12⋊9(C2×F5), C30⋊3(C4⋊C4), C20⋊2(C2×C12), (C2×C20)⋊3C12, (C4×D5)⋊4C12, D5.1(C6×D4), D5.1(C6×Q8), (D5×C12)⋊11C4, (C6×D5).56D4, (C6×D5).14Q8, D10.4(C3×Q8), (C6×Dic5)⋊18C4, Dic5⋊7(C2×C12), (C2×Dic5)⋊8C12, D10.10(C3×D4), (C22×F5).2C6, C22.18(C6×F5), C6.49(C22×F5), D10.15(C2×C12), C30.87(C22×C4), C10.5(C22×C12), (C6×D5).68C23, D10.9(C22×C6), (C6×F5).14C22, (D5×C12).133C22, C5⋊(C6×C4⋊C4), C10⋊(C3×C4⋊C4), D5⋊(C3×C4⋊C4), C15⋊4(C2×C4⋊C4), C2.6(C2×C6×F5), (C2×C4)⋊3(C3×F5), (C2×C6×F5).5C2, (C2×C4×D5).14C6, (C3×D5)⋊5(C4⋊C4), (D5×C2×C12).35C2, (C2×F5).1(C2×C6), (C2×C6).61(C2×F5), (C3×D5).6(C2×Q8), (C2×C30).60(C2×C4), (C4×D5).31(C2×C6), (C6×D5).64(C2×C4), (C3×D5).10(C2×D4), (C2×C10).17(C2×C12), (C3×Dic5)⋊28(C2×C4), (D5×C2×C6).150C22, (C22×D5).39(C2×C6), SmallGroup(480,1051)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×C4⋊F5
G = < a,b,c,d | a6=b4=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >
Subgroups: 584 in 184 conjugacy classes, 92 normal (32 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, C23, D5, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C4⋊C4, C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C2×C12, C2×C12, C22×C6, C3×D5, C3×D5, C30, C30, C2×C4⋊C4, C4×D5, C2×Dic5, C2×C20, C2×F5, C2×F5, C22×D5, C3×C4⋊C4, C22×C12, C3×Dic5, C60, C3×F5, C6×D5, C6×D5, C2×C30, C4⋊F5, C2×C4×D5, C22×F5, C6×C4⋊C4, D5×C12, C6×Dic5, C2×C60, C6×F5, C6×F5, D5×C2×C6, C2×C4⋊F5, C3×C4⋊F5, D5×C2×C12, C2×C6×F5, C6×C4⋊F5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C23, C12, C2×C6, C4⋊C4, C22×C4, C2×D4, C2×Q8, F5, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C4⋊C4, C2×F5, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, C3×F5, C4⋊F5, C22×F5, C6×C4⋊C4, C6×F5, C2×C4⋊F5, C3×C4⋊F5, C2×C6×F5, C6×C4⋊F5
►On 120 points
Generators in S120
(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 61 97 104)(2 62 98 105)(3 63 99 106)(4 64 100 107)(5 65 101 108)(6 66 102 103)(7 109 44 17)(8 110 45 18)(9 111 46 13)(10 112 47 14)(11 113 48 15)(12 114 43 16)(19 51 32 25)(20 52 33 26)(21 53 34 27)(22 54 35 28)(23 49 36 29)(24 50 31 30)(37 58 85 81)(38 59 86 82)(39 60 87 83)(40 55 88 84)(41 56 89 79)(42 57 90 80)(67 119 76 93)(68 120 77 94)(69 115 78 95)(70 116 73 96)(71 117 74 91)(72 118 75 92)
(1 95 41 29 109)(2 96 42 30 110)(3 91 37 25 111)(4 92 38 26 112)(5 93 39 27 113)(6 94 40 28 114)(7 104 78 79 36)(8 105 73 80 31)(9 106 74 81 32)(10 107 75 82 33)(11 108 76 83 34)(12 103 77 84 35)(13 99 117 85 51)(14 100 118 86 52)(15 101 119 87 53)(16 102 120 88 54)(17 97 115 89 49)(18 98 116 90 50)(19 46 63 71 58)(20 47 64 72 59)(21 48 65 67 60)(22 43 66 68 55)(23 44 61 69 56)(24 45 62 70 57)
(1 64)(2 65)(3 66)(4 61)(5 62)(6 63)(7 52 78 86)(8 53 73 87)(9 54 74 88)(10 49 75 89)(11 50 76 90)(12 51 77 85)(13 35 117 84)(14 36 118 79)(15 31 119 80)(16 32 120 81)(17 33 115 82)(18 34 116 83)(19 94 58 114)(20 95 59 109)(21 96 60 110)(22 91 55 111)(23 92 56 112)(24 93 57 113)(25 68 37 43)(26 69 38 44)(27 70 39 45)(28 71 40 46)(29 72 41 47)(30 67 42 48)(97 107)(98 108)(99 103)(100 104)(101 105)(102 106)
G:=sub<Sym(120)| (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), (1,61,97,104)(2,62,98,105)(3,63,99,106)(4,64,100,107)(5,65,101,108)(6,66,102,103)(7,109,44,17)(8,110,45,18)(9,111,46,13)(10,112,47,14)(11,113,48,15)(12,114,43,16)(19,51,32,25)(20,52,33,26)(21,53,34,27)(22,54,35,28)(23,49,36,29)(24,50,31,30)(37,58,85,81)(38,59,86,82)(39,60,87,83)(40,55,88,84)(41,56,89,79)(42,57,90,80)(67,119,76,93)(68,120,77,94)(69,115,78,95)(70,116,73,96)(71,117,74,91)(72,118,75,92), (1,95,41,29,109)(2,96,42,30,110)(3,91,37,25,111)(4,92,38,26,112)(5,93,39,27,113)(6,94,40,28,114)(7,104,78,79,36)(8,105,73,80,31)(9,106,74,81,32)(10,107,75,82,33)(11,108,76,83,34)(12,103,77,84,35)(13,99,117,85,51)(14,100,118,86,52)(15,101,119,87,53)(16,102,120,88,54)(17,97,115,89,49)(18,98,116,90,50)(19,46,63,71,58)(20,47,64,72,59)(21,48,65,67,60)(22,43,66,68,55)(23,44,61,69,56)(24,45,62,70,57), (1,64)(2,65)(3,66)(4,61)(5,62)(6,63)(7,52,78,86)(8,53,73,87)(9,54,74,88)(10,49,75,89)(11,50,76,90)(12,51,77,85)(13,35,117,84)(14,36,118,79)(15,31,119,80)(16,32,120,81)(17,33,115,82)(18,34,116,83)(19,94,58,114)(20,95,59,109)(21,96,60,110)(22,91,55,111)(23,92,56,112)(24,93,57,113)(25,68,37,43)(26,69,38,44)(27,70,39,45)(28,71,40,46)(29,72,41,47)(30,67,42,48)(97,107)(98,108)(99,103)(100,104)(101,105)(102,106)>;
G:=Group( (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), (1,61,97,104)(2,62,98,105)(3,63,99,106)(4,64,100,107)(5,65,101,108)(6,66,102,103)(7,109,44,17)(8,110,45,18)(9,111,46,13)(10,112,47,14)(11,113,48,15)(12,114,43,16)(19,51,32,25)(20,52,33,26)(21,53,34,27)(22,54,35,28)(23,49,36,29)(24,50,31,30)(37,58,85,81)(38,59,86,82)(39,60,87,83)(40,55,88,84)(41,56,89,79)(42,57,90,80)(67,119,76,93)(68,120,77,94)(69,115,78,95)(70,116,73,96)(71,117,74,91)(72,118,75,92), (1,95,41,29,109)(2,96,42,30,110)(3,91,37,25,111)(4,92,38,26,112)(5,93,39,27,113)(6,94,40,28,114)(7,104,78,79,36)(8,105,73,80,31)(9,106,74,81,32)(10,107,75,82,33)(11,108,76,83,34)(12,103,77,84,35)(13,99,117,85,51)(14,100,118,86,52)(15,101,119,87,53)(16,102,120,88,54)(17,97,115,89,49)(18,98,116,90,50)(19,46,63,71,58)(20,47,64,72,59)(21,48,65,67,60)(22,43,66,68,55)(23,44,61,69,56)(24,45,62,70,57), (1,64)(2,65)(3,66)(4,61)(5,62)(6,63)(7,52,78,86)(8,53,73,87)(9,54,74,88)(10,49,75,89)(11,50,76,90)(12,51,77,85)(13,35,117,84)(14,36,118,79)(15,31,119,80)(16,32,120,81)(17,33,115,82)(18,34,116,83)(19,94,58,114)(20,95,59,109)(21,96,60,110)(22,91,55,111)(23,92,56,112)(24,93,57,113)(25,68,37,43)(26,69,38,44)(27,70,39,45)(28,71,40,46)(29,72,41,47)(30,67,42,48)(97,107)(98,108)(99,103)(100,104)(101,105)(102,106) );
G=PermutationGroup([[(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)], [(1,61,97,104),(2,62,98,105),(3,63,99,106),(4,64,100,107),(5,65,101,108),(6,66,102,103),(7,109,44,17),(8,110,45,18),(9,111,46,13),(10,112,47,14),(11,113,48,15),(12,114,43,16),(19,51,32,25),(20,52,33,26),(21,53,34,27),(22,54,35,28),(23,49,36,29),(24,50,31,30),(37,58,85,81),(38,59,86,82),(39,60,87,83),(40,55,88,84),(41,56,89,79),(42,57,90,80),(67,119,76,93),(68,120,77,94),(69,115,78,95),(70,116,73,96),(71,117,74,91),(72,118,75,92)], [(1,95,41,29,109),(2,96,42,30,110),(3,91,37,25,111),(4,92,38,26,112),(5,93,39,27,113),(6,94,40,28,114),(7,104,78,79,36),(8,105,73,80,31),(9,106,74,81,32),(10,107,75,82,33),(11,108,76,83,34),(12,103,77,84,35),(13,99,117,85,51),(14,100,118,86,52),(15,101,119,87,53),(16,102,120,88,54),(17,97,115,89,49),(18,98,116,90,50),(19,46,63,71,58),(20,47,64,72,59),(21,48,65,67,60),(22,43,66,68,55),(23,44,61,69,56),(24,45,62,70,57)], [(1,64),(2,65),(3,66),(4,61),(5,62),(6,63),(7,52,78,86),(8,53,73,87),(9,54,74,88),(10,49,75,89),(11,50,76,90),(12,51,77,85),(13,35,117,84),(14,36,118,79),(15,31,119,80),(16,32,120,81),(17,33,115,82),(18,34,116,83),(19,94,58,114),(20,95,59,109),(21,96,60,110),(22,91,55,111),(23,92,56,112),(24,93,57,113),(25,68,37,43),(26,69,38,44),(27,70,39,45),(28,71,40,46),(29,72,41,47),(30,67,42,48),(97,107),(98,108),(99,103),(100,104),(101,105),(102,106)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | ··· | 4L | 5 | 6A | ··· | 6F | 6G | ··· | 6N | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 12E | ··· | 12X | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | ··· | 4 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 1 | 2 | 2 | 10 | ··· | 10 | 4 | 1 | ··· | 1 | 5 | ··· | 5 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | C12 | D4 | Q8 | C3×D4 | C3×Q8 | F5 | C2×F5 | C2×F5 | C3×F5 | C4⋊F5 | C6×F5 | C6×F5 | C3×C4⋊F5 |
| kernel | C6×C4⋊F5 | C3×C4⋊F5 | D5×C2×C12 | C2×C6×F5 | C2×C4⋊F5 | D5×C12 | C6×Dic5 | C2×C60 | C4⋊F5 | C2×C4×D5 | C22×F5 | C4×D5 | C2×Dic5 | C2×C20 | C6×D5 | C6×D5 | D10 | D10 | C2×C12 | C12 | C2×C6 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 2 | 4 | 2 | 2 | 8 | 2 | 4 | 8 | 4 | 4 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 |
Matrix representation of C6×C4⋊F5 ►in GL8(𝔽61)
| 48 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 48 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 47 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 47 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 47 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 47 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 47 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 47 |
| 42 | 56 | 0 | 0 | 0 | 0 | 0 | 0 |
| 48 | 19 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 42 | 56 | 0 | 0 | 0 | 0 |
| 0 | 0 | 48 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 60 | 60 | 60 | 60 |
| 26 | 55 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 35 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 60 | 60 | 60 |
G:=sub<GL(8,GF(61))| [48,0,0,0,0,0,0,0,0,48,0,0,0,0,0,0,0,0,47,0,0,0,0,0,0,0,0,47,0,0,0,0,0,0,0,0,47,0,0,0,0,0,0,0,0,47,0,0,0,0,0,0,0,0,47,0,0,0,0,0,0,0,0,47],[42,48,0,0,0,0,0,0,56,19,0,0,0,0,0,0,0,0,42,48,0,0,0,0,0,0,56,19,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,60,0,0,0,0,1,0,0,60,0,0,0,0,0,1,0,60,0,0,0,0,0,0,1,60],[26,1,0,0,0,0,0,0,55,35,0,0,0,0,0,0,0,0,35,60,0,0,0,0,0,0,6,26,0,0,0,0,0,0,0,0,1,0,0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,60,0,0,0,0,0,1,0,60] >;
C6×C4⋊F5 in GAP, Magma, Sage, TeX
C_6\times C_4\rtimes F_5
% in TeX
G:=Group("C6xC4:F5"); // GroupNames label
G:=SmallGroup(480,1051);
// by ID
G=gap.SmallGroup(480,1051);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,1094,268,9414,818]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^4=c^5=d^4=1,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