metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊5F5, C120⋊5C4, C40⋊4Dic3, C30.9C42, C8⋊3(C3⋊F5), C15⋊C8⋊5C4, C6.2(C4×F5), C3⋊1(C8⋊F5), C15⋊4(C8⋊C4), C5⋊2(C24⋊C4), C60.55(C2×C4), (C8×D5).10S3, (C4×D5).94D6, C5⋊2C8⋊8Dic3, C12.57(C2×F5), (D5×C24).15C2, D10.13(C4×S3), C10.9(C4×Dic3), C60.C4.5C2, D5.2(C8⋊S3), Dic5.18(C4×S3), C20.17(C2×Dic3), (C3×D5).3M4(2), (D5×C12).122C22, C2.3(C4×C3⋊F5), (C2×C3⋊F5).3C4, (C4×C3⋊F5).5C2, C4.16(C2×C3⋊F5), (C3×C5⋊2C8)⋊14C4, (C6×D5).40(C2×C4), (C3×Dic5).48(C2×C4), SmallGroup(480,297)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊F5
G = < a,b,c | a24=b5=c4=1, ab=ba, cac-1=a5, cbc-1=b3 >
Subgroups: 364 in 80 conjugacy classes, 37 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, D5, C10, Dic3, C12, C12, C2×C6, C15, C42, C2×C8, Dic5, C20, F5, D10, C3⋊C8, C24, C24, C2×Dic3, C2×C12, C3×D5, C30, C8⋊C4, C5⋊2C8, C40, C5⋊C8, C4×D5, C2×F5, C2×C3⋊C8, C4×Dic3, C2×C24, C3×Dic5, C60, C3⋊F5, C6×D5, C8×D5, D5⋊C8, C4×F5, C24⋊C4, C3×C5⋊2C8, C120, C15⋊C8, D5×C12, C2×C3⋊F5, C8⋊F5, D5×C24, C60.C4, C4×C3⋊F5, C24⋊F5
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C42, M4(2), F5, C4×S3, C2×Dic3, C8⋊C4, C2×F5, C8⋊S3, C4×Dic3, C3⋊F5, C4×F5, C24⋊C4, C2×C3⋊F5, C8⋊F5, C4×C3⋊F5, C24⋊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 53 78 104 32)(2 54 79 105 33)(3 55 80 106 34)(4 56 81 107 35)(5 57 82 108 36)(6 58 83 109 37)(7 59 84 110 38)(8 60 85 111 39)(9 61 86 112 40)(10 62 87 113 41)(11 63 88 114 42)(12 64 89 115 43)(13 65 90 116 44)(14 66 91 117 45)(15 67 92 118 46)(16 68 93 119 47)(17 69 94 120 48)(18 70 95 97 25)(19 71 96 98 26)(20 72 73 99 27)(21 49 74 100 28)(22 50 75 101 29)(23 51 76 102 30)(24 52 77 103 31)
(2 6)(3 11)(4 16)(5 21)(8 12)(9 17)(10 22)(14 18)(15 23)(20 24)(25 117 70 91)(26 98 71 96)(27 103 72 77)(28 108 49 82)(29 113 50 87)(30 118 51 92)(31 99 52 73)(32 104 53 78)(33 109 54 83)(34 114 55 88)(35 119 56 93)(36 100 57 74)(37 105 58 79)(38 110 59 84)(39 115 60 89)(40 120 61 94)(41 101 62 75)(42 106 63 80)(43 111 64 85)(44 116 65 90)(45 97 66 95)(46 102 67 76)(47 107 68 81)(48 112 69 86)
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), (1,53,78,104,32)(2,54,79,105,33)(3,55,80,106,34)(4,56,81,107,35)(5,57,82,108,36)(6,58,83,109,37)(7,59,84,110,38)(8,60,85,111,39)(9,61,86,112,40)(10,62,87,113,41)(11,63,88,114,42)(12,64,89,115,43)(13,65,90,116,44)(14,66,91,117,45)(15,67,92,118,46)(16,68,93,119,47)(17,69,94,120,48)(18,70,95,97,25)(19,71,96,98,26)(20,72,73,99,27)(21,49,74,100,28)(22,50,75,101,29)(23,51,76,102,30)(24,52,77,103,31), (2,6)(3,11)(4,16)(5,21)(8,12)(9,17)(10,22)(14,18)(15,23)(20,24)(25,117,70,91)(26,98,71,96)(27,103,72,77)(28,108,49,82)(29,113,50,87)(30,118,51,92)(31,99,52,73)(32,104,53,78)(33,109,54,83)(34,114,55,88)(35,119,56,93)(36,100,57,74)(37,105,58,79)(38,110,59,84)(39,115,60,89)(40,120,61,94)(41,101,62,75)(42,106,63,80)(43,111,64,85)(44,116,65,90)(45,97,66,95)(46,102,67,76)(47,107,68,81)(48,112,69,86)>;
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), (1,53,78,104,32)(2,54,79,105,33)(3,55,80,106,34)(4,56,81,107,35)(5,57,82,108,36)(6,58,83,109,37)(7,59,84,110,38)(8,60,85,111,39)(9,61,86,112,40)(10,62,87,113,41)(11,63,88,114,42)(12,64,89,115,43)(13,65,90,116,44)(14,66,91,117,45)(15,67,92,118,46)(16,68,93,119,47)(17,69,94,120,48)(18,70,95,97,25)(19,71,96,98,26)(20,72,73,99,27)(21,49,74,100,28)(22,50,75,101,29)(23,51,76,102,30)(24,52,77,103,31), (2,6)(3,11)(4,16)(5,21)(8,12)(9,17)(10,22)(14,18)(15,23)(20,24)(25,117,70,91)(26,98,71,96)(27,103,72,77)(28,108,49,82)(29,113,50,87)(30,118,51,92)(31,99,52,73)(32,104,53,78)(33,109,54,83)(34,114,55,88)(35,119,56,93)(36,100,57,74)(37,105,58,79)(38,110,59,84)(39,115,60,89)(40,120,61,94)(41,101,62,75)(42,106,63,80)(43,111,64,85)(44,116,65,90)(45,97,66,95)(46,102,67,76)(47,107,68,81)(48,112,69,86) );
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)], [(1,53,78,104,32),(2,54,79,105,33),(3,55,80,106,34),(4,56,81,107,35),(5,57,82,108,36),(6,58,83,109,37),(7,59,84,110,38),(8,60,85,111,39),(9,61,86,112,40),(10,62,87,113,41),(11,63,88,114,42),(12,64,89,115,43),(13,65,90,116,44),(14,66,91,117,45),(15,67,92,118,46),(16,68,93,119,47),(17,69,94,120,48),(18,70,95,97,25),(19,71,96,98,26),(20,72,73,99,27),(21,49,74,100,28),(22,50,75,101,29),(23,51,76,102,30),(24,52,77,103,31)], [(2,6),(3,11),(4,16),(5,21),(8,12),(9,17),(10,22),(14,18),(15,23),(20,24),(25,117,70,91),(26,98,71,96),(27,103,72,77),(28,108,49,82),(29,113,50,87),(30,118,51,92),(31,99,52,73),(32,104,53,78),(33,109,54,83),(34,114,55,88),(35,119,56,93),(36,100,57,74),(37,105,58,79),(38,110,59,84),(39,115,60,89),(40,120,61,94),(41,101,62,75),(42,106,63,80),(43,111,64,85),(44,116,65,90),(45,97,66,95),(46,102,67,76),(47,107,68,81),(48,112,69,86)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10 | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | 30A | 30B | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 5 | 5 | 2 | 1 | 1 | 5 | 5 | 30 | 30 | 30 | 30 | 4 | 2 | 10 | 10 | 2 | 2 | 10 | 10 | 30 | 30 | 30 | 30 | 4 | 2 | 2 | 10 | 10 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | - | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | Dic3 | Dic3 | D6 | M4(2) | C4×S3 | C4×S3 | C8⋊S3 | F5 | C2×F5 | C3⋊F5 | C4×F5 | C2×C3⋊F5 | C8⋊F5 | C4×C3⋊F5 | C24⋊F5 |
| kernel | C24⋊F5 | D5×C24 | C60.C4 | C4×C3⋊F5 | C3×C5⋊2C8 | C120 | C15⋊C8 | C2×C3⋊F5 | C8×D5 | C5⋊2C8 | C40 | C4×D5 | C3×D5 | Dic5 | D10 | D5 | C24 | C12 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C24⋊F5 ►in GL4(𝔽241) generated by
| 131 | 213 | 0 | 28 |
| 0 | 103 | 213 | 28 |
| 28 | 213 | 103 | 0 |
| 28 | 0 | 213 | 131 |
| 0 | 0 | 0 | 240 |
| 1 | 0 | 0 | 240 |
| 0 | 1 | 0 | 240 |
| 0 | 0 | 1 | 240 |
| 240 | 0 | 1 | 0 |
| 0 | 0 | 1 | 240 |
| 0 | 240 | 1 | 0 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(241))| [131,0,28,28,213,103,213,0,0,213,103,213,28,28,0,131],[0,1,0,0,0,0,1,0,0,0,0,1,240,240,240,240],[240,0,0,0,0,0,240,0,1,1,1,1,0,240,0,0] >;
C24⋊F5 in GAP, Magma, Sage, TeX
C_{24}\rtimes F_5 % in TeX
G:=Group("C24:F5"); // GroupNames label
G:=SmallGroup(480,297);
// by ID
G=gap.SmallGroup(480,297);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,64,80,2693,14118,4724]);
// Polycyclic
G:=Group<a,b,c|a^24=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=b^3>;
// generators/relations