direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: F5×C24, C120⋊7C4, C40⋊3C12, C30.13C42, C5⋊C8⋊3C12, C15⋊5(C4×C8), C5⋊1(C4×C24), D5.(C2×C24), C5⋊2C8⋊7C12, D5⋊C8.3C6, (C8×D5).9C6, (C6×F5).4C4, (C4×F5).3C6, C4.15(C6×F5), C2.1(C12×F5), C6.13(C4×F5), C10.1(C4×C12), C60.68(C2×C4), (C2×F5).2C12, (C12×F5).6C2, C12.68(C2×F5), C20.15(C2×C12), (D5×C24).20C2, D10.5(C2×C12), Dic5.7(C2×C12), (D5×C12).135C22, (C3×C5⋊C8)⋊7C4, (C3×C5⋊2C8)⋊17C4, (C3×D5⋊C8).6C2, (C3×D5).4(C2×C8), (C4×D5).32(C2×C6), (C6×D5).43(C2×C4), (C3×Dic5).51(C2×C4), SmallGroup(480,271)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — F5×C24 |
Generators and relations for F5×C24
G = < a,b,c | a24=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >
Subgroups: 232 in 88 conjugacy classes, 52 normal (32 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, D5, C10, C12, C12, C2×C6, C15, C42, C2×C8, Dic5, C20, F5, D10, C24, C24, C2×C12, C3×D5, C30, C4×C8, C5⋊2C8, C40, C5⋊C8, C4×D5, C2×F5, C4×C12, C2×C24, C3×Dic5, C60, C3×F5, C6×D5, C8×D5, D5⋊C8, C4×F5, C4×C24, C3×C5⋊2C8, C120, C3×C5⋊C8, D5×C12, C6×F5, C8×F5, D5×C24, C3×D5⋊C8, C12×F5, F5×C24
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C12, C2×C6, C42, C2×C8, F5, C24, C2×C12, C4×C8, C2×F5, C4×C12, C2×C24, C3×F5, C4×F5, C4×C24, C6×F5, C8×F5, C12×F5, F5×C24
►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 71 98 73 43)(2 72 99 74 44)(3 49 100 75 45)(4 50 101 76 46)(5 51 102 77 47)(6 52 103 78 48)(7 53 104 79 25)(8 54 105 80 26)(9 55 106 81 27)(10 56 107 82 28)(11 57 108 83 29)(12 58 109 84 30)(13 59 110 85 31)(14 60 111 86 32)(15 61 112 87 33)(16 62 113 88 34)(17 63 114 89 35)(18 64 115 90 36)(19 65 116 91 37)(20 66 117 92 38)(21 67 118 93 39)(22 68 119 94 40)(23 69 120 95 41)(24 70 97 96 42)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 91 53 116)(26 92 54 117)(27 93 55 118)(28 94 56 119)(29 95 57 120)(30 96 58 97)(31 73 59 98)(32 74 60 99)(33 75 61 100)(34 76 62 101)(35 77 63 102)(36 78 64 103)(37 79 65 104)(38 80 66 105)(39 81 67 106)(40 82 68 107)(41 83 69 108)(42 84 70 109)(43 85 71 110)(44 86 72 111)(45 87 49 112)(46 88 50 113)(47 89 51 114)(48 90 52 115)
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), (1,71,98,73,43)(2,72,99,74,44)(3,49,100,75,45)(4,50,101,76,46)(5,51,102,77,47)(6,52,103,78,48)(7,53,104,79,25)(8,54,105,80,26)(9,55,106,81,27)(10,56,107,82,28)(11,57,108,83,29)(12,58,109,84,30)(13,59,110,85,31)(14,60,111,86,32)(15,61,112,87,33)(16,62,113,88,34)(17,63,114,89,35)(18,64,115,90,36)(19,65,116,91,37)(20,66,117,92,38)(21,67,118,93,39)(22,68,119,94,40)(23,69,120,95,41)(24,70,97,96,42), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,91,53,116)(26,92,54,117)(27,93,55,118)(28,94,56,119)(29,95,57,120)(30,96,58,97)(31,73,59,98)(32,74,60,99)(33,75,61,100)(34,76,62,101)(35,77,63,102)(36,78,64,103)(37,79,65,104)(38,80,66,105)(39,81,67,106)(40,82,68,107)(41,83,69,108)(42,84,70,109)(43,85,71,110)(44,86,72,111)(45,87,49,112)(46,88,50,113)(47,89,51,114)(48,90,52,115)>;
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), (1,71,98,73,43)(2,72,99,74,44)(3,49,100,75,45)(4,50,101,76,46)(5,51,102,77,47)(6,52,103,78,48)(7,53,104,79,25)(8,54,105,80,26)(9,55,106,81,27)(10,56,107,82,28)(11,57,108,83,29)(12,58,109,84,30)(13,59,110,85,31)(14,60,111,86,32)(15,61,112,87,33)(16,62,113,88,34)(17,63,114,89,35)(18,64,115,90,36)(19,65,116,91,37)(20,66,117,92,38)(21,67,118,93,39)(22,68,119,94,40)(23,69,120,95,41)(24,70,97,96,42), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,91,53,116)(26,92,54,117)(27,93,55,118)(28,94,56,119)(29,95,57,120)(30,96,58,97)(31,73,59,98)(32,74,60,99)(33,75,61,100)(34,76,62,101)(35,77,63,102)(36,78,64,103)(37,79,65,104)(38,80,66,105)(39,81,67,106)(40,82,68,107)(41,83,69,108)(42,84,70,109)(43,85,71,110)(44,86,72,111)(45,87,49,112)(46,88,50,113)(47,89,51,114)(48,90,52,115) );
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)], [(1,71,98,73,43),(2,72,99,74,44),(3,49,100,75,45),(4,50,101,76,46),(5,51,102,77,47),(6,52,103,78,48),(7,53,104,79,25),(8,54,105,80,26),(9,55,106,81,27),(10,56,107,82,28),(11,57,108,83,29),(12,58,109,84,30),(13,59,110,85,31),(14,60,111,86,32),(15,61,112,87,33),(16,62,113,88,34),(17,63,114,89,35),(18,64,115,90,36),(19,65,116,91,37),(20,66,117,92,38),(21,67,118,93,39),(22,68,119,94,40),(23,69,120,95,41),(24,70,97,96,42)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,91,53,116),(26,92,54,117),(27,93,55,118),(28,94,56,119),(29,95,57,120),(30,96,58,97),(31,73,59,98),(32,74,60,99),(33,75,61,100),(34,76,62,101),(35,77,63,102),(36,78,64,103),(37,79,65,104),(38,80,66,105),(39,81,67,106),(40,82,68,107),(41,83,69,108),(42,84,70,109),(43,85,71,110),(44,86,72,111),(45,87,49,112),(46,88,50,113),(47,89,51,114),(48,90,52,115)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | ··· | 4L | 5 | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 8C | 8D | 8E | ··· | 8P | 10 | 12A | 12B | 12C | 12D | 12E | ··· | 12X | 15A | 15B | 20A | 20B | 24A | ··· | 24H | 24I | ··· | 24AF | 30A | 30B | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | ··· | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 20 | 20 | 24 | ··· | 24 | 24 | ··· | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 1 | 1 | 5 | ··· | 5 | 4 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | 5 | ··· | 5 | 4 | 1 | 1 | 1 | 1 | 5 | ··· | 5 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 5 | ··· | 5 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | ||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C4 | C4 | C6 | C6 | C6 | C8 | C12 | C12 | C12 | C12 | C24 | F5 | C2×F5 | C3×F5 | C4×F5 | C6×F5 | C8×F5 | C12×F5 | F5×C24 |
| kernel | F5×C24 | D5×C24 | C3×D5⋊C8 | C12×F5 | C8×F5 | C3×C5⋊2C8 | C120 | C3×C5⋊C8 | C6×F5 | C8×D5 | D5⋊C8 | C4×F5 | C3×F5 | C5⋊2C8 | C40 | C5⋊C8 | C2×F5 | F5 | C24 | C12 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 16 | 4 | 4 | 8 | 8 | 32 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of F5×C24 ►in GL5(𝔽241)
| 121 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 240 | 240 | 240 | 240 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(5,GF(241))| [121,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,240,1,0,0,0,240,0,1,0,0,240,0,0,1,0,240,0,0,0],[177,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,0] >;
F5×C24 in GAP, Magma, Sage, TeX
F_5\times C_{24} % in TeX
G:=Group("F5xC24"); // GroupNames label
G:=SmallGroup(480,271);
// by ID
G=gap.SmallGroup(480,271);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,176,102,9414,1595]);
// Polycyclic
G:=Group<a,b,c|a^24=b^5=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations