Copied to
clipboard

G = C2xDic3xF5order 480 = 25·3·5

Direct product of C2, Dic3 and F5

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C2xDic3xF5, C30:C42, C6:2(C4xF5), (C6xF5):2C4, (C3xD5):C42, C15:2(C2xC42), D5:1(C4xDic3), (D5xDic3):4C4, C10:1(C4xDic3), (C2xF5).12D6, Dic15:5(C2xC4), (C2xDic15):5C4, (C10xDic3):4C4, D10.24(C4xS3), D5.(C22xDic3), (C22xF5).3S3, C22.17(S3xF5), C6.17(C22xF5), C30.17(C22xC4), D10.8(C2xDic3), (C22xD5).73D6, (C6xD5).29C23, (C6xF5).11C22, D10.32(C22xS3), (D5xDic3).16C22, C3:3(C2xC4xF5), C3:F5:2(C2xC4), (C2xC3:F5):2C4, C5:1(C2xC4xDic3), C2.3(C2xS3xF5), D5.2(S3xC2xC4), (C2xC6xF5).2C2, C10.17(S3xC2xC4), (C3xF5):2(C2xC4), (C2xC6).18(C2xF5), (C2xC30).12(C2xC4), (C2xC10).14(C4xS3), (C22xC3:F5).2C2, (C5xDic3):5(C2xC4), (C6xD5).22(C2xC4), (C2xD5xDic3).11C2, (D5xC2xC6).66C22, (C2xC3:F5).11C22, (C3xD5).2(C22xC4), SmallGroup(480,998)

Series: Derived Chief Lower central Upper central

C1C15 — C2xDic3xF5
C1C5C15C3xD5C6xD5C6xF5Dic3xF5 — C2xDic3xF5
C15 — C2xDic3xF5
C1C22

Generators and relations for C2xDic3xF5
 G = < a,b,c,d,e | a2=b6=d5=e4=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 884 in 216 conjugacy classes, 94 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, C6, C6, C6, C2xC4, C23, D5, D5, C10, C10, Dic3, Dic3, C12, C2xC6, C2xC6, C15, C42, C22xC4, Dic5, C20, F5, F5, D10, D10, C2xC10, C2xDic3, C2xDic3, C2xC12, C22xC6, C3xD5, C3xD5, C30, C30, C2xC42, C4xD5, C2xDic5, C2xC20, C2xF5, C2xF5, C22xD5, C4xDic3, C22xDic3, C22xC12, C5xDic3, Dic15, C3xF5, C3:F5, C6xD5, C6xD5, C2xC30, C4xF5, C2xC4xD5, C22xF5, C22xF5, C2xC4xDic3, D5xDic3, C10xDic3, C2xDic15, C6xF5, C2xC3:F5, D5xC2xC6, C2xC4xF5, Dic3xF5, C2xD5xDic3, C2xC6xF5, C22xC3:F5, C2xDic3xF5
Quotients: C1, C2, C4, C22, S3, C2xC4, C23, Dic3, D6, C42, C22xC4, F5, C4xS3, C2xDic3, C22xS3, C2xC42, C2xF5, C4xDic3, S3xC2xC4, C22xDic3, C4xF5, C22xF5, C2xC4xDic3, S3xF5, C2xC4xF5, Dic3xF5, C2xS3xF5, C2xDic3xF5

Smallest permutation representation of C2xDic3xF5
On 120 points
Generators in S120
(1 58)(2 59)(3 60)(4 55)(5 56)(6 57)(7 112)(8 113)(9 114)(10 109)(11 110)(12 111)(13 108)(14 103)(15 104)(16 105)(17 106)(18 107)(19 98)(20 99)(21 100)(22 101)(23 102)(24 97)(25 94)(26 95)(27 96)(28 91)(29 92)(30 93)(31 72)(32 67)(33 68)(34 69)(35 70)(36 71)(37 62)(38 63)(39 64)(40 65)(41 66)(42 61)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)
(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 89 4 86)(2 88 5 85)(3 87 6 90)(7 84 10 81)(8 83 11 80)(9 82 12 79)(13 78 16 75)(14 77 17 74)(15 76 18 73)(19 72 22 69)(20 71 23 68)(21 70 24 67)(25 66 28 63)(26 65 29 62)(27 64 30 61)(31 101 34 98)(32 100 35 97)(33 99 36 102)(37 95 40 92)(38 94 41 91)(39 93 42 96)(43 104 46 107)(44 103 47 106)(45 108 48 105)(49 114 52 111)(50 113 53 110)(51 112 54 109)(55 116 58 119)(56 115 59 118)(57 120 60 117)
(1 47 32 51 39)(2 48 33 52 40)(3 43 34 53 41)(4 44 35 54 42)(5 45 36 49 37)(6 46 31 50 38)(7 30 119 17 21)(8 25 120 18 22)(9 26 115 13 23)(10 27 116 14 24)(11 28 117 15 19)(12 29 118 16 20)(55 74 70 84 61)(56 75 71 79 62)(57 76 72 80 63)(58 77 67 81 64)(59 78 68 82 65)(60 73 69 83 66)(85 108 102 114 95)(86 103 97 109 96)(87 104 98 110 91)(88 105 99 111 92)(89 106 100 112 93)(90 107 101 113 94)
(1 119 4 116)(2 120 5 117)(3 115 6 118)(7 44 24 39)(8 45 19 40)(9 46 20 41)(10 47 21 42)(11 48 22 37)(12 43 23 38)(13 31 29 53)(14 32 30 54)(15 33 25 49)(16 34 26 50)(17 35 27 51)(18 36 28 52)(55 86 58 89)(56 87 59 90)(57 88 60 85)(61 109 77 100)(62 110 78 101)(63 111 73 102)(64 112 74 97)(65 113 75 98)(66 114 76 99)(67 93 84 103)(68 94 79 104)(69 95 80 105)(70 96 81 106)(71 91 82 107)(72 92 83 108)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E···4L4M···4X 5 6A6B6C6D6E6F6G10A10B10C12A···12H 15 20A20B20C20D30A30B30C
order12222222344444···44···45666666610101012···121520202020303030
size11115555233335···515···1542221010101044410···10812121212888

60 irreducible representations

dim11111111112222224444888
type++++++-++++++-+
imageC1C2C2C2C2C4C4C4C4C4S3Dic3D6D6C4xS3C4xS3F5C2xF5C2xF5C4xF5S3xF5Dic3xF5C2xS3xF5
kernelC2xDic3xF5Dic3xF5C2xD5xDic3C2xC6xF5C22xC3:F5D5xDic3C10xDic3C2xDic15C6xF5C2xC3:F5C22xF5C2xF5C2xF5C22xD5D10C2xC10C2xDic3Dic3C2xC6C6C22C2C2
# reps14111422881421621214121

Matrix representation of C2xDic3xF5 in GL6(F61)

100000
010000
0060000
0006000
0000600
0000060
,
0600000
110000
001000
000100
000010
000001
,
1100000
50500000
0060000
0006000
0000600
0000060
,
100000
010000
0060606060
001000
000100
000010
,
100000
010000
001000
000001
000100
0060606060

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[0,1,0,0,0,0,60,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,50,0,0,0,0,0,50,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,60,0,1,0,0,0,60,0,0,1,0,0,60,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,60,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,1,0,60] >;

C2xDic3xF5 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_3\times F_5
% in TeX

G:=Group("C2xDic3xF5");
// GroupNames label

G:=SmallGroup(480,998);
// by ID

G=gap.SmallGroup(480,998);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,1356,9414,2379]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^6=d^5=e^4=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
// generators/relations

׿
x
:
Z
F
o
wr
Q
<