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G = C24⋊F5order 480 = 25·3·5

5th semidirect product of C24 and F5 acting via F5/D5=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C245F5, C1205C4, C404Dic3, C30.9C42, C83(C3⋊F5), C15⋊C85C4, C6.2(C4×F5), C31(C8⋊F5), C154(C8⋊C4), C52(C24⋊C4), C60.55(C2×C4), (C8×D5).10S3, (C4×D5).94D6, C52C88Dic3, 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×C52C8)⋊14C4, (C6×D5).40(C2×C4), (C3×Dic5).48(C2×C4), SmallGroup(480,297)

Series: Derived Chief Lower central Upper central

C1C30 — C24⋊F5
C1C5C15C30C6×D5D5×C12C4×C3⋊F5 — C24⋊F5
C15C30 — C24⋊F5
C1C4C8

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, C52C8, 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×C52C8, 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

Smallest permutation representation of 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)
(1 112 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 2A2B2C 3 4A4B4C4D4E4F4G4H 5 6A6B6C8A8B8C8D8E8F8G8H 10 12A12B12C12D15A15B20A20B24A24B24C24D24E24F24G24H30A30B40A40B40C40D60A60B60C60D120A···120H
order1222344444444566688888888101212121215152020242424242424242430304040404060606060120···120
size11552115530303030421010221010303030304221010444422221010101044444444444···4

60 irreducible representations

dim111111112222222244444444
type+++++--+++
imageC1C2C2C2C4C4C4C4S3Dic3Dic3D6M4(2)C4×S3C4×S3C8⋊S3F5C2×F5C3⋊F5C4×F5C2×C3⋊F5C8⋊F5C4×C3⋊F5C24⋊F5
kernelC24⋊F5D5×C24C60.C4C4×C3⋊F5C3×C52C8C120C15⋊C8C2×C3⋊F5C8×D5C52C8C40C4×D5C3×D5Dic5D10D5C24C12C8C6C4C3C2C1
# reps111122441111422811222448

Matrix representation of C24⋊F5 in GL4(𝔽241) generated by

131213028
010321328
282131030
280213131
,
000240
100240
010240
001240
,
240010
001240
024010
0010
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

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