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G = C5×Q83Dic3order 480 = 25·3·5

Direct product of C5 and Q83Dic3

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

Aliases: C5×Q83Dic3, C60.241D4, C1523C4≀C2, (C3×D4)⋊2C20, (C3×Q8)⋊2C20, (D4×C15)⋊14C4, C12.9(C2×C20), (Q8×C15)⋊14C4, Q83(C5×Dic3), (C5×D4)⋊8Dic3, D42(C5×Dic3), C12.56(C5×D4), (C2×C30).90D4, C60.179(C2×C4), (C5×Q8)⋊11Dic3, (C4×Dic3)⋊2C10, (C2×C20).352D6, C4.Dic34C10, C4.3(C10×Dic3), (Dic3×C20)⋊14C2, C20.53(C2×Dic3), C20.124(C3⋊D4), (C2×C60).349C22, C30.122(C22⋊C4), C10.38(C6.D4), C33(C5×C4≀C2), (C2×C6).3(C5×D4), C4○D4.3(C5×S3), (C5×C4○D4).8S3, C4.31(C5×C3⋊D4), (C2×C4).40(S3×C10), (C3×C4○D4).1C10, (C15×C4○D4).7C2, C6.18(C5×C22⋊C4), C22.3(C5×C3⋊D4), (C2×C12).19(C2×C10), (C5×C4.Dic3)⋊16C2, C2.8(C5×C6.D4), (C2×C10).39(C3⋊D4), SmallGroup(480,156)

Series: Derived Chief Lower central Upper central

C1C12 — C5×Q83Dic3
C1C3C6C12C2×C12C2×C60C5×C4.Dic3 — C5×Q83Dic3
C3C6C12 — C5×Q83Dic3
C1C20C2×C20C5×C4○D4

Generators and relations for C5×Q83Dic3
 G = < a,b,c,d,e | a5=b4=d6=1, c2=b2, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b2c, ece-1=b-1c, ede-1=d-1 >

Subgroups: 180 in 88 conjugacy classes, 42 normal (all characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C5, C6, C6 [×2], C8, C2×C4, C2×C4 [×2], D4, D4, Q8, C10, C10 [×2], Dic3 [×2], C12 [×2], C12, C2×C6, C2×C6, C15, C42, M4(2), C4○D4, C20 [×2], C20 [×3], C2×C10, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C30, C30 [×2], C4≀C2, C40, C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C4.Dic3, C4×Dic3, C3×C4○D4, C5×Dic3 [×2], C60 [×2], C60, C2×C30, C2×C30, C4×C20, C5×M4(2), C5×C4○D4, Q83Dic3, C5×C3⋊C8, C10×Dic3, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, C5×C4≀C2, C5×C4.Dic3, Dic3×C20, C15×C4○D4, C5×Q83Dic3
Quotients: C1, C2 [×3], C4 [×2], C22, C5, S3, C2×C4, D4 [×2], C10 [×3], Dic3 [×2], D6, C22⋊C4, C20 [×2], C2×C10, C2×Dic3, C3⋊D4 [×2], C5×S3, C4≀C2, C2×C20, C5×D4 [×2], C6.D4, C5×Dic3 [×2], S3×C10, C5×C22⋊C4, Q83Dic3, C10×Dic3, C5×C3⋊D4 [×2], C5×C4≀C2, C5×C6.D4, C5×Q83Dic3

Smallest permutation representation of C5×Q83Dic3
On 120 points
Generators in S120
(1 49 37 25 13)(2 50 38 26 14)(3 51 39 27 15)(4 52 40 28 16)(5 53 41 29 17)(6 54 42 30 18)(7 55 43 31 19)(8 56 44 32 20)(9 57 45 33 21)(10 58 46 34 22)(11 59 47 35 23)(12 60 48 36 24)(61 109 97 85 76)(62 110 98 86 77)(63 111 99 87 78)(64 112 100 88 73)(65 113 101 89 74)(66 114 102 90 75)(67 115 103 91 79)(68 116 104 92 80)(69 117 105 93 81)(70 118 106 94 82)(71 119 107 95 83)(72 120 108 96 84)
(1 10 4 7)(2 11 5 8)(3 12 6 9)(13 22 16 19)(14 23 17 20)(15 24 18 21)(25 34 28 31)(26 35 29 32)(27 36 30 33)(37 46 40 43)(38 47 41 44)(39 48 42 45)(49 58 52 55)(50 59 53 56)(51 60 54 57)(61 67 64 70)(62 68 65 71)(63 69 66 72)(73 82 76 79)(74 83 77 80)(75 84 78 81)(85 91 88 94)(86 92 89 95)(87 93 90 96)(97 103 100 106)(98 104 101 107)(99 105 102 108)(109 115 112 118)(110 116 113 119)(111 117 114 120)
(1 67 4 70)(2 71 5 68)(3 69 6 72)(7 64 10 61)(8 62 11 65)(9 66 12 63)(13 79 16 82)(14 83 17 80)(15 81 18 84)(19 73 22 76)(20 77 23 74)(21 75 24 78)(25 91 28 94)(26 95 29 92)(27 93 30 96)(31 88 34 85)(32 86 35 89)(33 90 36 87)(37 103 40 106)(38 107 41 104)(39 105 42 108)(43 100 46 97)(44 98 47 101)(45 102 48 99)(49 115 52 118)(50 119 53 116)(51 117 54 120)(55 112 58 109)(56 110 59 113)(57 114 60 111)
(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 4)(2 6)(3 5)(7 10)(8 12)(9 11)(13 16)(14 18)(15 17)(19 22)(20 24)(21 23)(25 28)(26 30)(27 29)(31 34)(32 36)(33 35)(37 40)(38 42)(39 41)(43 46)(44 48)(45 47)(49 52)(50 54)(51 53)(55 58)(56 60)(57 59)(61 67 64 70)(62 72 65 69)(63 71 66 68)(73 82 76 79)(74 81 77 84)(75 80 78 83)(85 91 88 94)(86 96 89 93)(87 95 90 92)(97 103 100 106)(98 108 101 105)(99 107 102 104)(109 115 112 118)(110 120 113 117)(111 119 114 116)

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

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

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

120 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H5A5B5C5D6A6B6C6D8A8B10A10B10C10D10E10F10G10H10I10J10K10L12A12B12C12D12E15A15B15C15D20A···20H20I20J20K20L20M20N20O20P20Q···20AF30A30B30C30D30E···30P40A···40H60A···60H60I···60T
order1222344444444555566668810101010101010101010101012121212121515151520···20202020202020202020···203030303030···3040···4060···6060···60
size11242112466661111244412121111222244442244422221···1222244446···622224···412···122···24···4

120 irreducible representations

dim11111111111122222222222222222244
type++++++++--
imageC1C2C2C2C4C4C5C10C10C10C20C20S3D4D4D6Dic3Dic3C3⋊D4C3⋊D4C5×S3C4≀C2C5×D4C5×D4S3×C10C5×Dic3C5×Dic3C5×C3⋊D4C5×C3⋊D4C5×C4≀C2Q83Dic3C5×Q83Dic3
kernelC5×Q83Dic3C5×C4.Dic3Dic3×C20C15×C4○D4D4×C15Q8×C15Q83Dic3C4.Dic3C4×Dic3C3×C4○D4C3×D4C3×Q8C5×C4○D4C60C2×C30C2×C20C5×D4C5×Q8C20C2×C10C4○D4C15C12C2×C6C2×C4D4Q8C4C22C3C5C1
# reps111122444488111111224444444881628

Matrix representation of C5×Q83Dic3 in GL4(𝔽241) generated by

91000
09100
0010
0001
,
240000
024000
00640
0043177
,
7014000
10117100
0018549
0023656
,
24024000
1000
0010
00140240
,
1000
24024000
002400
0072177
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,1,0,0,0,0,1],[240,0,0,0,0,240,0,0,0,0,64,43,0,0,0,177],[70,101,0,0,140,171,0,0,0,0,185,236,0,0,49,56],[240,1,0,0,240,0,0,0,0,0,1,140,0,0,0,240],[1,240,0,0,0,240,0,0,0,0,240,72,0,0,0,177] >;

C5×Q83Dic3 in GAP, Magma, Sage, TeX

C_5\times Q_8\rtimes_3{\rm Dic}_3
% in TeX

G:=Group("C5xQ8:3Dic3");
// GroupNames label

G:=SmallGroup(480,156);
// by ID

G=gap.SmallGroup(480,156);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,140,589,136,4204,2111,102,15686]);
// Polycyclic

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

׿
×
𝔽