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G = F5×C2×C12order 480 = 25·3·5

Direct product of C2×C12 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — F5×C2×C12
 Chief series C1 — C5 — C10 — D10 — C6×D5 — C6×F5 — C2×C6×F5 — F5×C2×C12
 Lower central C5 — F5×C2×C12
 Upper central C1 — C2×C12

Generators and relations for F5×C2×C12
G = < a,b,c,d | a2=b12=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 584 in 216 conjugacy classes, 124 normal (28 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, C23, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C42, C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C2×C12, C2×C12, C22×C6, C3×D5, C30, C30, C2×C42, C4×D5, C2×Dic5, C2×C20, C2×F5, C22×D5, C4×C12, C22×C12, C3×Dic5, C60, C3×F5, C6×D5, C6×D5, C2×C30, C4×F5, C2×C4×D5, C22×F5, C2×C4×C12, D5×C12, C6×Dic5, C2×C60, C6×F5, D5×C2×C6, C2×C4×F5, C12×F5, D5×C2×C12, C2×C6×F5, F5×C2×C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C42, C22×C4, F5, C2×C12, C22×C6, C2×C42, C2×F5, C4×C12, C22×C12, C3×F5, C4×F5, C22×F5, C2×C4×C12, C6×F5, C2×C4×F5, C12×F5, C2×C6×F5, F5×C2×C12

Smallest permutation representation of F5×C2×C12
On 120 points
Generators in S120
(1 48)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 120)(14 109)(15 110)(16 111)(17 112)(18 113)(19 114)(20 115)(21 116)(22 117)(23 118)(24 119)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(49 101)(50 102)(51 103)(52 104)(53 105)(54 106)(55 107)(56 108)(57 97)(58 98)(59 99)(60 100)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)
(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)
(1 119 63 56 73)(2 120 64 57 74)(3 109 65 58 75)(4 110 66 59 76)(5 111 67 60 77)(6 112 68 49 78)(7 113 69 50 79)(8 114 70 51 80)(9 115 71 52 81)(10 116 72 53 82)(11 117 61 54 83)(12 118 62 55 84)(13 34 97 86 37)(14 35 98 87 38)(15 36 99 88 39)(16 25 100 89 40)(17 26 101 90 41)(18 27 102 91 42)(19 28 103 92 43)(20 29 104 93 44)(21 30 105 94 45)(22 31 106 95 46)(23 32 107 96 47)(24 33 108 85 48)
(1 48)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 64 86 57)(14 65 87 58)(15 66 88 59)(16 67 89 60)(17 68 90 49)(18 69 91 50)(19 70 92 51)(20 71 93 52)(21 72 94 53)(22 61 95 54)(23 62 96 55)(24 63 85 56)(25 77 100 111)(26 78 101 112)(27 79 102 113)(28 80 103 114)(29 81 104 115)(30 82 105 116)(31 83 106 117)(32 84 107 118)(33 73 108 119)(34 74 97 120)(35 75 98 109)(36 76 99 110)

G:=sub<Sym(120)| (1,48)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,120)(14,109)(15,110)(16,111)(17,112)(18,113)(19,114)(20,115)(21,116)(22,117)(23,118)(24,119)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(49,101)(50,102)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,97)(58,98)(59,99)(60,100)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (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), (1,119,63,56,73)(2,120,64,57,74)(3,109,65,58,75)(4,110,66,59,76)(5,111,67,60,77)(6,112,68,49,78)(7,113,69,50,79)(8,114,70,51,80)(9,115,71,52,81)(10,116,72,53,82)(11,117,61,54,83)(12,118,62,55,84)(13,34,97,86,37)(14,35,98,87,38)(15,36,99,88,39)(16,25,100,89,40)(17,26,101,90,41)(18,27,102,91,42)(19,28,103,92,43)(20,29,104,93,44)(21,30,105,94,45)(22,31,106,95,46)(23,32,107,96,47)(24,33,108,85,48), (1,48)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,64,86,57)(14,65,87,58)(15,66,88,59)(16,67,89,60)(17,68,90,49)(18,69,91,50)(19,70,92,51)(20,71,93,52)(21,72,94,53)(22,61,95,54)(23,62,96,55)(24,63,85,56)(25,77,100,111)(26,78,101,112)(27,79,102,113)(28,80,103,114)(29,81,104,115)(30,82,105,116)(31,83,106,117)(32,84,107,118)(33,73,108,119)(34,74,97,120)(35,75,98,109)(36,76,99,110)>;

G:=Group( (1,48)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,120)(14,109)(15,110)(16,111)(17,112)(18,113)(19,114)(20,115)(21,116)(22,117)(23,118)(24,119)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(49,101)(50,102)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,97)(58,98)(59,99)(60,100)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (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), (1,119,63,56,73)(2,120,64,57,74)(3,109,65,58,75)(4,110,66,59,76)(5,111,67,60,77)(6,112,68,49,78)(7,113,69,50,79)(8,114,70,51,80)(9,115,71,52,81)(10,116,72,53,82)(11,117,61,54,83)(12,118,62,55,84)(13,34,97,86,37)(14,35,98,87,38)(15,36,99,88,39)(16,25,100,89,40)(17,26,101,90,41)(18,27,102,91,42)(19,28,103,92,43)(20,29,104,93,44)(21,30,105,94,45)(22,31,106,95,46)(23,32,107,96,47)(24,33,108,85,48), (1,48)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,64,86,57)(14,65,87,58)(15,66,88,59)(16,67,89,60)(17,68,90,49)(18,69,91,50)(19,70,92,51)(20,71,93,52)(21,72,94,53)(22,61,95,54)(23,62,96,55)(24,63,85,56)(25,77,100,111)(26,78,101,112)(27,79,102,113)(28,80,103,114)(29,81,104,115)(30,82,105,116)(31,83,106,117)(32,84,107,118)(33,73,108,119)(34,74,97,120)(35,75,98,109)(36,76,99,110) );

G=PermutationGroup([[(1,48),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,43),(9,44),(10,45),(11,46),(12,47),(13,120),(14,109),(15,110),(16,111),(17,112),(18,113),(19,114),(20,115),(21,116),(22,117),(23,118),(24,119),(25,67),(26,68),(27,69),(28,70),(29,71),(30,72),(31,61),(32,62),(33,63),(34,64),(35,65),(36,66),(49,101),(50,102),(51,103),(52,104),(53,105),(54,106),(55,107),(56,108),(57,97),(58,98),(59,99),(60,100),(73,85),(74,86),(75,87),(76,88),(77,89),(78,90),(79,91),(80,92),(81,93),(82,94),(83,95),(84,96)], [(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)], [(1,119,63,56,73),(2,120,64,57,74),(3,109,65,58,75),(4,110,66,59,76),(5,111,67,60,77),(6,112,68,49,78),(7,113,69,50,79),(8,114,70,51,80),(9,115,71,52,81),(10,116,72,53,82),(11,117,61,54,83),(12,118,62,55,84),(13,34,97,86,37),(14,35,98,87,38),(15,36,99,88,39),(16,25,100,89,40),(17,26,101,90,41),(18,27,102,91,42),(19,28,103,92,43),(20,29,104,93,44),(21,30,105,94,45),(22,31,106,95,46),(23,32,107,96,47),(24,33,108,85,48)], [(1,48),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,43),(9,44),(10,45),(11,46),(12,47),(13,64,86,57),(14,65,87,58),(15,66,88,59),(16,67,89,60),(17,68,90,49),(18,69,91,50),(19,70,92,51),(20,71,93,52),(21,72,94,53),(22,61,95,54),(23,62,96,55),(24,63,85,56),(25,77,100,111),(26,78,101,112),(27,79,102,113),(28,80,103,114),(29,81,104,115),(30,82,105,116),(31,83,106,117),(32,84,107,118),(33,73,108,119),(34,74,97,120),(35,75,98,109),(36,76,99,110)]])

120 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E ··· 4X 5 6A ··· 6F 6G ··· 6N 10A 10B 10C 12A ··· 12H 12I ··· 12AV 15A 15B 20A 20B 20C 20D 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 2 2 3 3 4 4 4 4 4 ··· 4 5 6 ··· 6 6 ··· 6 10 10 10 12 ··· 12 12 ··· 12 15 15 20 20 20 20 30 ··· 30 60 ··· 60 size 1 1 1 1 5 5 5 5 1 1 1 1 1 1 5 ··· 5 4 1 ··· 1 5 ··· 5 4 4 4 1 ··· 1 5 ··· 5 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 4 4 4 4 4 4 4 4 type + + + + + + + image C1 C2 C2 C2 C3 C4 C4 C4 C4 C6 C6 C6 C12 C12 C12 C12 F5 C2×F5 C2×F5 C3×F5 C4×F5 C6×F5 C6×F5 C12×F5 kernel F5×C2×C12 C12×F5 D5×C2×C12 C2×C6×F5 C2×C4×F5 D5×C12 C6×Dic5 C2×C60 C6×F5 C4×F5 C2×C4×D5 C22×F5 C4×D5 C2×Dic5 C2×C20 C2×F5 C2×C12 C12 C2×C6 C2×C4 C6 C4 C22 C2 # reps 1 4 1 2 2 4 2 2 16 8 2 4 8 4 4 32 1 2 1 2 4 4 2 8

Matrix representation of F5×C2×C12 in GL5(𝔽61)

 60 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 11 0 0 0 0 0 21 0 0 0 0 0 21 0 0 0 0 0 21 0 0 0 0 0 21
,
 1 0 0 0 0 0 60 60 60 60 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0
,
 50 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(61))| [60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[11,0,0,0,0,0,21,0,0,0,0,0,21,0,0,0,0,0,21,0,0,0,0,0,21],[1,0,0,0,0,0,60,1,0,0,0,60,0,1,0,0,60,0,0,1,0,60,0,0,0],[50,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,0] >;

F5×C2×C12 in GAP, Magma, Sage, TeX

F_5\times C_2\times C_{12}
% in TeX

G:=Group("F5xC2xC12");
// GroupNames label

G:=SmallGroup(480,1050);
// by ID

G=gap.SmallGroup(480,1050);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,268,9414,818]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

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