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G = C3×D42Dic5order 480 = 25·3·5

Direct product of C3 and D42Dic5

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

Aliases: C3×D42Dic5, C60.234D4, C1522C4≀C2, (C5×D4)⋊5C12, (C5×Q8)⋊8C12, (D4×C15)⋊11C4, (Q8×C15)⋊11C4, Q83(C3×Dic5), (C4×Dic5)⋊2C6, (C3×D4)⋊5Dic5, (C3×Q8)⋊5Dic5, D42(C3×Dic5), C20.56(C3×D4), (C2×C30).80D4, C4.Dic54C6, C4.3(C6×Dic5), C20.30(C2×C12), C60.164(C2×C4), (C12×Dic5)⋊14C2, (C2×C12).356D10, C12.32(C2×Dic5), C12.124(C5⋊D4), (C2×C60).283C22, C6.27(C23.D5), C30.115(C22⋊C4), C55(C3×C4≀C2), C4○D4.3(C3×D5), (C3×C4○D4).4D5, (C5×C4○D4).5C6, (C2×C4).35(C6×D5), (C2×C10).3(C3×D4), C4.31(C3×C5⋊D4), (C2×C20).19(C2×C6), (C15×C4○D4).4C2, C22.3(C3×C5⋊D4), C2.8(C3×C23.D5), (C2×C6).39(C5⋊D4), C10.29(C3×C22⋊C4), (C3×C4.Dic5)⋊16C2, SmallGroup(480,115)

Series: Derived Chief Lower central Upper central

C1C20 — C3×D42Dic5
C1C5C10C20C2×C20C2×C60C3×C4.Dic5 — C3×D42Dic5
C5C10C20 — C3×D42Dic5
C1C12C2×C12C3×C4○D4

Generators and relations for C3×D42Dic5
 G = < a,b,c,d,e | a3=b4=d10=1, c2=b2, e2=d5, 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: 224 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], C12 [×2], C12 [×3], C2×C6, C2×C6, C15, C42, M4(2), C4○D4, Dic5 [×2], C20 [×2], C20, C2×C10, C2×C10, C24, C2×C12, C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C30, C30 [×2], C4≀C2, C52C8, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C4×C12, C3×M4(2), C3×C4○D4, C3×Dic5 [×2], C60 [×2], C60, C2×C30, C2×C30, C4.Dic5, C4×Dic5, C5×C4○D4, C3×C4≀C2, C3×C52C8, C6×Dic5, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, D42Dic5, C3×C4.Dic5, C12×Dic5, C15×C4○D4, C3×D42Dic5
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4 [×2], D5, C12 [×2], C2×C6, C22⋊C4, Dic5 [×2], D10, C2×C12, C3×D4 [×2], C3×D5, C4≀C2, C2×Dic5, C5⋊D4 [×2], C3×C22⋊C4, C3×Dic5 [×2], C6×D5, C23.D5, C3×C4≀C2, C6×Dic5, C3×C5⋊D4 [×2], D42Dic5, C3×C23.D5, C3×D42Dic5

Smallest permutation representation of C3×D42Dic5
On 120 points
Generators in S120
(1 41 21)(2 42 22)(3 43 23)(4 44 24)(5 45 25)(6 46 26)(7 47 27)(8 48 28)(9 49 29)(10 50 30)(11 51 31)(12 52 32)(13 53 33)(14 54 34)(15 55 35)(16 56 36)(17 57 37)(18 58 38)(19 59 39)(20 60 40)(61 101 81)(62 102 82)(63 103 83)(64 104 84)(65 105 85)(66 106 86)(67 107 87)(68 108 88)(69 109 89)(70 110 90)(71 111 91)(72 112 92)(73 113 93)(74 114 94)(75 115 95)(76 116 96)(77 117 97)(78 118 98)(79 119 99)(80 120 100)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 80)(81 91 86 96)(82 92 87 97)(83 93 88 98)(84 94 89 99)(85 95 90 100)(101 111 106 116)(102 112 107 117)(103 113 108 118)(104 114 109 119)(105 115 110 120)
(1 71 6 76)(2 77 7 72)(3 73 8 78)(4 79 9 74)(5 75 10 80)(11 66 16 61)(12 62 17 67)(13 68 18 63)(14 64 19 69)(15 70 20 65)(21 91 26 96)(22 97 27 92)(23 93 28 98)(24 99 29 94)(25 95 30 100)(31 86 36 81)(32 82 37 87)(33 88 38 83)(34 84 39 89)(35 90 40 85)(41 111 46 116)(42 117 47 112)(43 113 48 118)(44 119 49 114)(45 115 50 120)(51 106 56 101)(52 102 57 107)(53 108 58 103)(54 104 59 109)(55 110 60 105)
(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 6)(2 10)(3 9)(4 8)(5 7)(11 16)(12 20)(13 19)(14 18)(15 17)(21 26)(22 30)(23 29)(24 28)(25 27)(31 36)(32 40)(33 39)(34 38)(35 37)(41 46)(42 50)(43 49)(44 48)(45 47)(51 56)(52 60)(53 59)(54 58)(55 57)(61 71 66 76)(62 80 67 75)(63 79 68 74)(64 78 69 73)(65 77 70 72)(81 91 86 96)(82 100 87 95)(83 99 88 94)(84 98 89 93)(85 97 90 92)(101 111 106 116)(102 120 107 115)(103 119 108 114)(104 118 109 113)(105 117 110 112)

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

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

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

102 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D4E4F4G4H5A5B6A6B6C6D6E6F8A8B10A10B10C···10H12A12B12C12D12E12F12G12H12I···12P15A15B15C15D20A20B20C20D20E···20J24A24B24C24D30A30B30C30D30E···30P60A···60H60I···60T
order122233444444445566666688101010···10121212121212121212···12151515152020202020···20242424243030303030···3060···6060···60
size112411112410101010221122442020224···41111224410···10222222224···42020202022224···42···24···4

102 irreducible representations

dim11111111111122222222222222222244
type++++++++--
imageC1C2C2C2C3C4C4C6C6C6C12C12D4D4D5D10Dic5Dic5C3×D4C3×D4C3×D5C4≀C2C5⋊D4C5⋊D4C6×D5C3×Dic5C3×Dic5C3×C4≀C2C3×C5⋊D4C3×C5⋊D4D42Dic5C3×D42Dic5
kernelC3×D42Dic5C3×C4.Dic5C12×Dic5C15×C4○D4D42Dic5D4×C15Q8×C15C4.Dic5C4×Dic5C5×C4○D4C5×D4C5×Q8C60C2×C30C3×C4○D4C2×C12C3×D4C3×Q8C20C2×C10C4○D4C15C12C2×C6C2×C4D4Q8C5C4C22C3C1
# reps11112222224411222222444444488848

Matrix representation of C3×D42Dic5 in GL6(𝔽241)

100000
010000
0015000
0001500
000010
000001
,
6400000
1771770000
00240000
00024000
000010
000001
,
641280000
01770000
001000
00124000
00002400
00000240
,
100000
2402400000
001000
000100
00000240
0000151
,
24000000
891770000
00123900
00024000
000051190
0000240190

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[64,177,0,0,0,0,0,177,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[64,0,0,0,0,0,128,177,0,0,0,0,0,0,1,1,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,240,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,240,51],[240,89,0,0,0,0,0,177,0,0,0,0,0,0,1,0,0,0,0,0,239,240,0,0,0,0,0,0,51,240,0,0,0,0,190,190] >;

C3×D42Dic5 in GAP, Magma, Sage, TeX

C_3\times D_4\rtimes_2{\rm Dic}_5
% in TeX

G:=Group("C3xD4:2Dic5");
// GroupNames label

G:=SmallGroup(480,115);
// by ID

G=gap.SmallGroup(480,115);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,136,2524,1271,102,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=d^10=1,c^2=b^2,e^2=d^5,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