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 [×2], C3, C4, C4 [×3], C22, C5, C6, C6 [×2], C8, C8 [×3], C2×C4 [×3], D5 [×2], C10, Dic3 [×2], C12, C12, C2×C6, C15, C42, C2×C8 [×2], Dic5, C20, F5 [×2], D10, C3⋊C8 [×2], C24, C24, C2×Dic3 [×2], C2×C12, C3×D5 [×2], C30, C8⋊C4, C5⋊2C8, C40, C5⋊C8 [×2], C4×D5, C2×F5 [×2], C2×C3⋊C8, C4×Dic3, C2×C24, C3×Dic5, C60, C3⋊F5 [×2], C6×D5, C8×D5, D5⋊C8, C4×F5, C24⋊C4, C3×C5⋊2C8, C120, C15⋊C8 [×2], D5×C12, C2×C3⋊F5 [×2], C8⋊F5, D5×C24, C60.C4, C4×C3⋊F5, C24⋊F5
Quotients: C1, C2 [×3], C4 [×6], C22, S3, C2×C4 [×3], Dic3 [×2], D6, C42, M4(2) [×2], F5, C4×S3 [×2], C2×Dic3, C8⋊C4, C2×F5, C8⋊S3 [×2], C4×Dic3, C3⋊F5, C4×F5, C24⋊C4, C2×C3⋊F5, C8⋊F5, C4×C3⋊F5, C24⋊F5
►On 120 points
(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)
(1 112 48 64 87)(2 113 25 65 88)(3 114 26 66 89)(4 115 27 67 90)(5 116 28 68 91)(6 117 29 69 92)(7 118 30 70 93)(8 119 31 71 94)(9 120 32 72 95)(10 97 33 49 96)(11 98 34 50 73)(12 99 35 51 74)(13 100 36 52 75)(14 101 37 53 76)(15 102 38 54 77)(16 103 39 55 78)(17 104 40 56 79)(18 105 41 57 80)(19 106 42 58 81)(20 107 43 59 82)(21 108 44 60 83)(22 109 45 61 84)(23 110 46 62 85)(24 111 47 63 86)
(2 6)(3 11)(4 16)(5 21)(8 12)(9 17)(10 22)(14 18)(15 23)(20 24)(25 92 65 117)(26 73 66 98)(27 78 67 103)(28 83 68 108)(29 88 69 113)(30 93 70 118)(31 74 71 99)(32 79 72 104)(33 84 49 109)(34 89 50 114)(35 94 51 119)(36 75 52 100)(37 80 53 105)(38 85 54 110)(39 90 55 115)(40 95 56 120)(41 76 57 101)(42 81 58 106)(43 86 59 111)(44 91 60 116)(45 96 61 97)(46 77 62 102)(47 82 63 107)(48 87 64 112)
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), (1,112,48,64,87)(2,113,25,65,88)(3,114,26,66,89)(4,115,27,67,90)(5,116,28,68,91)(6,117,29,69,92)(7,118,30,70,93)(8,119,31,71,94)(9,120,32,72,95)(10,97,33,49,96)(11,98,34,50,73)(12,99,35,51,74)(13,100,36,52,75)(14,101,37,53,76)(15,102,38,54,77)(16,103,39,55,78)(17,104,40,56,79)(18,105,41,57,80)(19,106,42,58,81)(20,107,43,59,82)(21,108,44,60,83)(22,109,45,61,84)(23,110,46,62,85)(24,111,47,63,86), (2,6)(3,11)(4,16)(5,21)(8,12)(9,17)(10,22)(14,18)(15,23)(20,24)(25,92,65,117)(26,73,66,98)(27,78,67,103)(28,83,68,108)(29,88,69,113)(30,93,70,118)(31,74,71,99)(32,79,72,104)(33,84,49,109)(34,89,50,114)(35,94,51,119)(36,75,52,100)(37,80,53,105)(38,85,54,110)(39,90,55,115)(40,95,56,120)(41,76,57,101)(42,81,58,106)(43,86,59,111)(44,91,60,116)(45,96,61,97)(46,77,62,102)(47,82,63,107)(48,87,64,112)>;
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), (1,112,48,64,87)(2,113,25,65,88)(3,114,26,66,89)(4,115,27,67,90)(5,116,28,68,91)(6,117,29,69,92)(7,118,30,70,93)(8,119,31,71,94)(9,120,32,72,95)(10,97,33,49,96)(11,98,34,50,73)(12,99,35,51,74)(13,100,36,52,75)(14,101,37,53,76)(15,102,38,54,77)(16,103,39,55,78)(17,104,40,56,79)(18,105,41,57,80)(19,106,42,58,81)(20,107,43,59,82)(21,108,44,60,83)(22,109,45,61,84)(23,110,46,62,85)(24,111,47,63,86), (2,6)(3,11)(4,16)(5,21)(8,12)(9,17)(10,22)(14,18)(15,23)(20,24)(25,92,65,117)(26,73,66,98)(27,78,67,103)(28,83,68,108)(29,88,69,113)(30,93,70,118)(31,74,71,99)(32,79,72,104)(33,84,49,109)(34,89,50,114)(35,94,51,119)(36,75,52,100)(37,80,53,105)(38,85,54,110)(39,90,55,115)(40,95,56,120)(41,76,57,101)(42,81,58,106)(43,86,59,111)(44,91,60,116)(45,96,61,97)(46,77,62,102)(47,82,63,107)(48,87,64,112) );
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)], [(1,112,48,64,87),(2,113,25,65,88),(3,114,26,66,89),(4,115,27,67,90),(5,116,28,68,91),(6,117,29,69,92),(7,118,30,70,93),(8,119,31,71,94),(9,120,32,72,95),(10,97,33,49,96),(11,98,34,50,73),(12,99,35,51,74),(13,100,36,52,75),(14,101,37,53,76),(15,102,38,54,77),(16,103,39,55,78),(17,104,40,56,79),(18,105,41,57,80),(19,106,42,58,81),(20,107,43,59,82),(21,108,44,60,83),(22,109,45,61,84),(23,110,46,62,85),(24,111,47,63,86)], [(2,6),(3,11),(4,16),(5,21),(8,12),(9,17),(10,22),(14,18),(15,23),(20,24),(25,92,65,117),(26,73,66,98),(27,78,67,103),(28,83,68,108),(29,88,69,113),(30,93,70,118),(31,74,71,99),(32,79,72,104),(33,84,49,109),(34,89,50,114),(35,94,51,119),(36,75,52,100),(37,80,53,105),(38,85,54,110),(39,90,55,115),(40,95,56,120),(41,76,57,101),(42,81,58,106),(43,86,59,111),(44,91,60,116),(45,96,61,97),(46,77,62,102),(47,82,63,107),(48,87,64,112)])
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