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G = D5×C6.D4order 480 = 25·3·5

Direct product of D5 and C6.D4

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

Aliases: D5×C6.D4, (C6×D5).68D4, C6.155(D4×D5), D107(C2×Dic3), C30.227(C2×D4), (C23×D5).4S3, C224(D5×Dic3), C23.40(S3×D5), (C2×Dic3)⋊12D10, (C22×D5)⋊6Dic3, (C22×C6).90D10, (C22×C10).44D6, C30.38D420C2, D10⋊Dic329C2, D10.42(C3⋊D4), C30.143(C22×C4), (C2×C30).189C23, (C2×Dic15)⋊9C22, (C10×Dic3)⋊5C22, (C22×D5).110D6, (C22×C30).51C22, C10.31(C22×Dic3), (D5×C2×C6)⋊6C4, C35(D5×C22⋊C4), C6.94(C2×C4×D5), (C2×C6)⋊14(C4×D5), C2.5(D5×C3⋊D4), (C2×C30)⋊14(C2×C4), (C6×D5)⋊27(C2×C4), C1511(C2×C22⋊C4), (C2×D5×Dic3)⋊13C2, C52(C2×C6.D4), C2.18(C2×D5×Dic3), (D5×C22×C6).2C2, C22.83(C2×S3×D5), (C2×C10)⋊8(C2×Dic3), C10.59(C2×C3⋊D4), (C3×D5)⋊3(C22⋊C4), (C5×C6.D4)⋊8C2, (D5×C2×C6).111C22, (C2×C6).201(C22×D5), (C2×C10).201(C22×S3), SmallGroup(480,623)

Series: Derived Chief Lower central Upper central

C1C30 — D5×C6.D4
C1C5C15C30C2×C30D5×C2×C6C2×D5×Dic3 — D5×C6.D4
C15C30 — D5×C6.D4
C1C22C23

Generators and relations for D5×C6.D4
 G = < a,b,c,d,e | a5=b2=c6=d4=1, e2=c3, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece-1=c-1, ede-1=c3d-1 >

Subgroups: 1212 in 264 conjugacy classes, 84 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×4], C22, C22 [×2], C22 [×20], C5, C6, C6 [×2], C6 [×8], C2×C4 [×8], C23, C23 [×10], D5 [×4], D5 [×2], C10, C10 [×2], C10 [×2], Dic3 [×4], C2×C6, C2×C6 [×2], C2×C6 [×20], C15, C22⋊C4 [×4], C22×C4 [×2], C24, Dic5 [×2], C20 [×2], D10 [×8], D10 [×10], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×Dic3 [×2], C2×Dic3 [×6], C22×C6, C22×C6 [×10], C3×D5 [×4], C3×D5 [×2], C30, C30 [×2], C30 [×2], C2×C22⋊C4, C4×D5 [×4], C2×Dic5 [×2], C2×C20 [×2], C22×D5 [×2], C22×D5 [×4], C22×D5 [×4], C22×C10, C6.D4, C6.D4 [×3], C22×Dic3 [×2], C23×C6, C5×Dic3 [×2], Dic15 [×2], C6×D5 [×8], C6×D5 [×10], C2×C30, C2×C30 [×2], C2×C30 [×2], D10⋊C4 [×2], C23.D5, C5×C22⋊C4, C2×C4×D5 [×2], C23×D5, C2×C6.D4, D5×Dic3 [×4], C10×Dic3 [×2], C2×Dic15 [×2], D5×C2×C6 [×2], D5×C2×C6 [×4], D5×C2×C6 [×4], C22×C30, D5×C22⋊C4, D10⋊Dic3 [×2], C5×C6.D4, C30.38D4, C2×D5×Dic3 [×2], D5×C22×C6, D5×C6.D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D5, Dic3 [×4], D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C2×Dic3 [×6], C3⋊D4 [×4], C22×S3, C2×C22⋊C4, C4×D5 [×2], C22×D5, C6.D4 [×4], C22×Dic3, C2×C3⋊D4 [×2], S3×D5, C2×C4×D5, D4×D5 [×2], C2×C6.D4, D5×Dic3 [×2], C2×S3×D5, D5×C22⋊C4, C2×D5×Dic3, D5×C3⋊D4 [×2], D5×C6.D4

Smallest permutation representation of D5×C6.D4
On 120 points
Generators in S120
(1 101 88 95 29)(2 102 89 96 30)(3 97 90 91 25)(4 98 85 92 26)(5 99 86 93 27)(6 100 87 94 28)(7 67 103 56 44)(8 68 104 57 45)(9 69 105 58 46)(10 70 106 59 47)(11 71 107 60 48)(12 72 108 55 43)(13 41 23 35 84)(14 42 24 36 79)(15 37 19 31 80)(16 38 20 32 81)(17 39 21 33 82)(18 40 22 34 83)(49 116 66 74 113)(50 117 61 75 114)(51 118 62 76 109)(52 119 63 77 110)(53 120 64 78 111)(54 115 65 73 112)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 25)(7 59)(8 60)(9 55)(10 56)(11 57)(12 58)(13 81)(14 82)(15 83)(16 84)(17 79)(18 80)(19 22)(20 23)(21 24)(31 40)(32 41)(33 42)(34 37)(35 38)(36 39)(43 46)(44 47)(45 48)(49 119)(50 120)(51 115)(52 116)(53 117)(54 118)(61 111)(62 112)(63 113)(64 114)(65 109)(66 110)(67 106)(68 107)(69 108)(70 103)(71 104)(72 105)(73 76)(74 77)(75 78)(85 88)(86 89)(87 90)(91 100)(92 101)(93 102)(94 97)(95 98)(96 99)
(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)
(1 116 13 104)(2 115 14 103)(3 120 15 108)(4 119 16 107)(5 118 17 106)(6 117 18 105)(7 96 112 36)(8 95 113 35)(9 94 114 34)(10 93 109 33)(11 92 110 32)(12 91 111 31)(19 43 90 78)(20 48 85 77)(21 47 86 76)(22 46 87 75)(23 45 88 74)(24 44 89 73)(25 53 80 72)(26 52 81 71)(27 51 82 70)(28 50 83 69)(29 49 84 68)(30 54 79 67)(37 55 97 64)(38 60 98 63)(39 59 99 62)(40 58 100 61)(41 57 101 66)(42 56 102 65)
(1 119 4 116)(2 118 5 115)(3 117 6 120)(7 36 10 33)(8 35 11 32)(9 34 12 31)(13 107 16 104)(14 106 17 103)(15 105 18 108)(19 46 22 43)(20 45 23 48)(21 44 24 47)(25 50 28 53)(26 49 29 52)(27 54 30 51)(37 58 40 55)(38 57 41 60)(39 56 42 59)(61 100 64 97)(62 99 65 102)(63 98 66 101)(67 79 70 82)(68 84 71 81)(69 83 72 80)(73 89 76 86)(74 88 77 85)(75 87 78 90)(91 114 94 111)(92 113 95 110)(93 112 96 109)

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

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

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

72 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K 3 4A4B4C4D4E4F4G4H5A5B6A···6G6H···6O10A···10F10G10H10I10J15A15B20A···20H30A···30N
order122222222222344444444556···66···610···1010101010151520···2030···30
size111122555510102666630303030222···210···102···244444412···124···4

72 irreducible representations

dim1111111222222222244444
type+++++++++-++++++-+
imageC1C2C2C2C2C2C4S3D4D5Dic3D6D6D10D10C3⋊D4C4×D5S3×D5D4×D5D5×Dic3C2×S3×D5D5×C3⋊D4
kernelD5×C6.D4D10⋊Dic3C5×C6.D4C30.38D4C2×D5×Dic3D5×C22×C6D5×C2×C6C23×D5C6×D5C6.D4C22×D5C22×D5C22×C10C2×Dic3C22×C6D10C2×C6C23C6C22C22C2
# reps1211218142421428824428

Matrix representation of D5×C6.D4 in GL4(𝔽61) generated by

1000
0100
0001
006017
,
1000
0100
00060
00600
,
476000
01300
00600
00060
,
13900
154800
00110
00011
,
13100
154800
00500
00050
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,0,60,0,0,1,17],[1,0,0,0,0,1,0,0,0,0,0,60,0,0,60,0],[47,0,0,0,60,13,0,0,0,0,60,0,0,0,0,60],[13,15,0,0,9,48,0,0,0,0,11,0,0,0,0,11],[13,15,0,0,1,48,0,0,0,0,50,0,0,0,0,50] >;

D5×C6.D4 in GAP, Magma, Sage, TeX

D_5\times C_6.D_4
% in TeX

G:=Group("D5xC6.D4");
// GroupNames label

G:=SmallGroup(480,623);
// by ID

G=gap.SmallGroup(480,623);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,219,1356,18822]);
// Polycyclic

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

׿
×
𝔽