Copied to
clipboard

G = D5×C24⋊C2order 480 = 25·3·5

Direct product of D5 and C24⋊C2

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

Aliases: D5×C24⋊C2, C4013D6, C2423D10, Dic67D10, C12013C22, D10.22D12, D12.18D10, Dic5.5D12, C60.116C23, D60.30C22, Dic3016C22, (C8×D5)⋊4S3, C811(S3×D5), C6.1(D4×D5), C31(D5×SD16), (D5×C24)⋊4C2, C52C823D6, C2.6(D5×D12), C30.1(C2×D4), C151(C2×SD16), (C3×D5)⋊1SD16, (D5×Dic6)⋊8C2, (C6×D5).40D4, (D5×D12).2C2, (C4×D5).76D6, C10.1(C2×D12), C24⋊D513C2, D12.D59C2, Dic6⋊D59C2, C20.64(C22×S3), (C3×Dic5).44D4, (C5×Dic6)⋊12C22, (C5×D12).20C22, (D5×C12).90C22, C12.139(C22×D5), C51(C2×C24⋊C2), C4.64(C2×S3×D5), (C5×C24⋊C2)⋊3C2, (C3×C52C8)⋊27C22, SmallGroup(480,323)

Series: Derived Chief Lower central Upper central

C1C60 — D5×C24⋊C2
C1C5C15C30C60D5×C12D5×D12 — D5×C24⋊C2
C15C30C60 — D5×C24⋊C2
C1C2C4C8

Generators and relations for D5×C24⋊C2
 G = < a,b,c,d | a5=b2=c24=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c11 >

Subgroups: 956 in 136 conjugacy classes, 44 normal (40 characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×3], C22 [×5], C5, S3 [×2], C6, C6 [×2], C8, C8, C2×C4 [×2], D4 [×3], Q8 [×3], C23, D5 [×2], D5, C10, C10, Dic3 [×2], C12, C12, D6 [×4], C2×C6, C15, C2×C8, SD16 [×4], C2×D4, C2×Q8, Dic5, Dic5, C20, C20, D10, D10 [×3], C2×C10, C24, C24, Dic6, Dic6 [×2], D12, D12 [×2], C2×Dic3, C2×C12, C22×S3, C5×S3, C3×D5 [×2], D15, C30, C2×SD16, C52C8, C40, Dic10 [×2], C4×D5, C4×D5, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, C24⋊C2, C24⋊C2 [×3], C2×C24, C2×Dic6, C2×D12, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5 [×2], C6×D5, S3×C10, D30, C8×D5, C40⋊C2, D4.D5, Q8⋊D5, C5×SD16, D4×D5, Q8×D5, C2×C24⋊C2, C3×C52C8, C120, D5×Dic3, C5⋊D12, C15⋊Q8, D5×C12, C5×Dic6, C5×D12, Dic30, D60, C2×S3×D5, D5×SD16, D12.D5, Dic6⋊D5, D5×C24, C5×C24⋊C2, C24⋊D5, D5×Dic6, D5×D12, D5×C24⋊C2
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], SD16 [×2], C2×D4, D10 [×3], D12 [×2], C22×S3, C2×SD16, C22×D5, C24⋊C2 [×2], C2×D12, S3×D5, D4×D5, C2×C24⋊C2, C2×S3×D5, D5×SD16, D5×D12, D5×C24⋊C2

Smallest permutation representation of D5×C24⋊C2
On 120 points
Generators in S120
(1 47 65 114 94)(2 48 66 115 95)(3 25 67 116 96)(4 26 68 117 73)(5 27 69 118 74)(6 28 70 119 75)(7 29 71 120 76)(8 30 72 97 77)(9 31 49 98 78)(10 32 50 99 79)(11 33 51 100 80)(12 34 52 101 81)(13 35 53 102 82)(14 36 54 103 83)(15 37 55 104 84)(16 38 56 105 85)(17 39 57 106 86)(18 40 58 107 87)(19 41 59 108 88)(20 42 60 109 89)(21 43 61 110 90)(22 44 62 111 91)(23 45 63 112 92)(24 46 64 113 93)
(1 94)(2 95)(3 96)(4 73)(5 74)(6 75)(7 76)(8 77)(9 78)(10 79)(11 80)(12 81)(13 82)(14 83)(15 84)(16 85)(17 86)(18 87)(19 88)(20 89)(21 90)(22 91)(23 92)(24 93)(25 116)(26 117)(27 118)(28 119)(29 120)(30 97)(31 98)(32 99)(33 100)(34 101)(35 102)(36 103)(37 104)(38 105)(39 106)(40 107)(41 108)(42 109)(43 110)(44 111)(45 112)(46 113)(47 114)(48 115)
(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)
(2 12)(3 23)(4 10)(5 21)(6 8)(7 19)(9 17)(11 15)(14 24)(16 22)(18 20)(25 45)(26 32)(27 43)(28 30)(29 41)(31 39)(33 37)(34 48)(36 46)(38 44)(40 42)(49 57)(50 68)(51 55)(52 66)(54 64)(56 62)(58 60)(59 71)(61 69)(63 67)(70 72)(73 79)(74 90)(75 77)(76 88)(78 86)(80 84)(81 95)(83 93)(85 91)(87 89)(92 96)(97 119)(98 106)(99 117)(100 104)(101 115)(103 113)(105 111)(107 109)(108 120)(110 118)(112 116)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D5A5B6A6B6C8A8B8C8D10A10B10C10D12A12B12C12D15A15B20A20B20C20D24A24B24C24D24E24F24G24H30A30B40A40B40C40D60A60B60C60D120A···120H
order122222344445566688881010101012121212151520202020242424242424242430304040404060606060120···120
size115512602210126022210102210102224242210104444242422221010101044444444444···4

60 irreducible representations

dim1111111122222222222222444444
type++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6SD16D10D10D10D12D12C24⋊C2S3×D5D4×D5C2×S3×D5D5×SD16D5×D12D5×C24⋊C2
kernelD5×C24⋊C2D12.D5Dic6⋊D5D5×C24C5×C24⋊C2C24⋊D5D5×Dic6D5×D12C8×D5C3×Dic5C6×D5C24⋊C2C52C8C40C4×D5C3×D5C24Dic6D12Dic5D10D5C8C6C4C3C2C1
# reps1111111111121114222228222448

Matrix representation of D5×C24⋊C2 in GL6(𝔽241)

100000
010000
001000
000100
00001891
00002400
,
100000
010000
001000
000100
00001189
00000240
,
203730000
20800000
00240100
00240000
00002400
00000240
,
11230000
02400000
000100
001000
000010
000001

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,189,240,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,189,240],[203,208,0,0,0,0,73,0,0,0,0,0,0,0,240,240,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,123,240,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

D5×C24⋊C2 in GAP, Magma, Sage, TeX

D_5\times C_{24}\rtimes C_2
% in TeX

G:=Group("D5xC24:C2");
// GroupNames label

G:=SmallGroup(480,323);
// by ID

G=gap.SmallGroup(480,323);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,135,58,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^24=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^11>;
// generators/relations

׿
×
𝔽