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G = F5×C24order 480 = 25·3·5

Direct product of C24 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — F5×C24
 Chief series C1 — C5 — C10 — C20 — C4×D5 — D5×C12 — C12×F5 — F5×C24
 Lower central C5 — F5×C24
 Upper central C1 — 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, C52C8, 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×C52C8, 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

Smallest permutation representation of 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

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