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G = C3×C23⋊Dic5order 480 = 25·3·5

Direct product of C3 and C23⋊Dic5

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

Aliases: C3×C23⋊Dic5, (C2×C20)⋊6C12, (C2×C60)⋊11C4, (C6×D4).8D5, C23⋊(C3×Dic5), (C22×C30)⋊3C4, (D4×C10).6C6, (C2×C12)⋊1Dic5, (C2×C30).79D4, C23.D52C6, C23.2(C6×D5), C1514(C23⋊C4), (C22×C10)⋊4C12, (D4×C30).16C2, (C22×C6)⋊1Dic5, (C22×C6).2D10, C22.3(C6×Dic5), C6.24(C23.D5), C30.112(C22⋊C4), (C22×C30).92C22, C54(C3×C23⋊C4), (C2×C4)⋊(C3×Dic5), (C2×D4).3(C3×D5), (C2×C10).2(C3×D4), C22.2(C3×C5⋊D4), (C2×C10).50(C2×C12), (C2×C30).186(C2×C4), C2.5(C3×C23.D5), (C2×C6).38(C5⋊D4), C10.26(C3×C22⋊C4), (C3×C23.D5)⋊18C2, (C2×C6).21(C2×Dic5), (C22×C10).11(C2×C6), SmallGroup(480,112)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C3×C23⋊Dic5
C1C5C10C2×C10C22×C10C22×C30C3×C23.D5 — C3×C23⋊Dic5
C5C10C2×C10 — C3×C23⋊Dic5
C1C6C22×C6C6×D4

Generators and relations for C3×C23⋊Dic5
 G = < a,b,c,d,e,f | a3=b2=c2=d2=e10=1, f2=e5, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=bd=db, fbf-1=bcd, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e-1 >

Subgroups: 320 in 104 conjugacy classes, 42 normal (30 characteristic)
C1, C2, C2 [×4], C3, C4 [×3], C22, C22 [×2], C22 [×3], C5, C6, C6 [×4], C2×C4, C2×C4 [×2], D4 [×2], C23 [×2], C10, C10 [×4], C12 [×3], C2×C6, C2×C6 [×2], C2×C6 [×3], C15, C22⋊C4 [×2], C2×D4, Dic5 [×2], C20, C2×C10, C2×C10 [×2], C2×C10 [×3], C2×C12, C2×C12 [×2], C3×D4 [×2], C22×C6 [×2], C30, C30 [×4], C23⋊C4, C2×Dic5 [×2], C2×C20, C5×D4 [×2], C22×C10 [×2], C3×C22⋊C4 [×2], C6×D4, C3×Dic5 [×2], C60, C2×C30, C2×C30 [×2], C2×C30 [×3], C23.D5 [×2], D4×C10, C3×C23⋊C4, C6×Dic5 [×2], C2×C60, D4×C15 [×2], C22×C30 [×2], C23⋊Dic5, C3×C23.D5 [×2], D4×C30, C3×C23⋊Dic5
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, C23⋊C4, C2×Dic5, C5⋊D4 [×2], C3×C22⋊C4, C3×Dic5 [×2], C6×D5, C23.D5, C3×C23⋊C4, C6×Dic5, C3×C5⋊D4 [×2], C23⋊Dic5, C3×C23.D5, C3×C23⋊Dic5

Smallest permutation representation of C3×C23⋊Dic5
On 120 points
Generators in S120
(1 54 34)(2 55 35)(3 56 36)(4 57 37)(5 58 38)(6 59 39)(7 60 40)(8 51 31)(9 52 32)(10 53 33)(11 110 90)(12 101 81)(13 102 82)(14 103 83)(15 104 84)(16 105 85)(17 106 86)(18 107 87)(19 108 88)(20 109 89)(21 66 41)(22 67 42)(23 68 43)(24 69 44)(25 70 45)(26 61 46)(27 62 47)(28 63 48)(29 64 49)(30 65 50)(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 82)(2 77)(3 84)(4 79)(5 86)(6 71)(7 88)(8 73)(9 90)(10 75)(11 52)(12 61)(13 54)(14 63)(15 56)(16 65)(17 58)(18 67)(19 60)(20 69)(21 80)(22 87)(23 72)(24 89)(25 74)(26 81)(27 76)(28 83)(29 78)(30 85)(31 93)(32 110)(33 95)(34 102)(35 97)(36 104)(37 99)(38 106)(39 91)(40 108)(41 100)(42 107)(43 92)(44 109)(45 94)(46 101)(47 96)(48 103)(49 98)(50 105)(51 113)(53 115)(55 117)(57 119)(59 111)(62 116)(64 118)(66 120)(68 112)(70 114)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 119)(12 120)(13 111)(14 112)(15 113)(16 114)(17 115)(18 116)(19 117)(20 118)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 82)(72 83)(73 84)(74 85)(75 86)(76 87)(77 88)(78 89)(79 90)(80 81)(91 102)(92 103)(93 104)(94 105)(95 106)(96 107)(97 108)(98 109)(99 110)(100 101)
(1 27)(2 28)(3 29)(4 30)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 114)(12 115)(13 116)(14 117)(15 118)(16 119)(17 120)(18 111)(19 112)(20 113)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(37 50)(38 41)(39 42)(40 43)(51 69)(52 70)(53 61)(54 62)(55 63)(56 64)(57 65)(58 66)(59 67)(60 68)(71 87)(72 88)(73 89)(74 90)(75 81)(76 82)(77 83)(78 84)(79 85)(80 86)(91 107)(92 108)(93 109)(94 110)(95 101)(96 102)(97 103)(98 104)(99 105)(100 106)
(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 82 6 87)(2 81 7 86)(3 90 8 85)(4 89 9 84)(5 88 10 83)(11 51 16 56)(12 60 17 55)(13 59 18 54)(14 58 19 53)(15 57 20 52)(21 72 26 77)(22 71 27 76)(23 80 28 75)(24 79 29 74)(25 78 30 73)(31 105 36 110)(32 104 37 109)(33 103 38 108)(34 102 39 107)(35 101 40 106)(41 92 46 97)(42 91 47 96)(43 100 48 95)(44 99 49 94)(45 98 50 93)(61 117 66 112)(62 116 67 111)(63 115 68 120)(64 114 69 119)(65 113 70 118)

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

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

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

93 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C4D4E5A5B6A6B6C···6H6I6J10A···10F10G···10N12A12B12C···12J15A15B15C15D20A20B20C20D30A···30L30M···30AB60A···60H
order122222334444455666···66610···1010···10121212···12151515152020202030···3030···3060···60
size1122241142020202022112···2442···24···44420···20222244442···24···44···4

93 irreducible representations

dim11111111112222222222224444
type+++++--++
imageC1C2C2C3C4C4C6C6C12C12D4D5Dic5Dic5D10C3×D4C3×D5C5⋊D4C3×Dic5C3×Dic5C6×D5C3×C5⋊D4C23⋊C4C3×C23⋊C4C23⋊Dic5C3×C23⋊Dic5
kernelC3×C23⋊Dic5C3×C23.D5D4×C30C23⋊Dic5C2×C60C22×C30C23.D5D4×C10C2×C20C22×C10C2×C30C6×D4C2×C12C22×C6C22×C6C2×C10C2×D4C2×C6C2×C4C23C23C22C15C5C3C1
# reps121222424422222448444161248

Matrix representation of C3×C23⋊Dic5 in GL4(𝔽61) generated by

47000
04700
00470
00047
,
0010
0001
1000
0100
,
474400
331400
004744
003314
,
60000
06000
00600
00060
,
223100
81400
003930
005347
,
002231
00639
17100
174400
G:=sub<GL(4,GF(61))| [47,0,0,0,0,47,0,0,0,0,47,0,0,0,0,47],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[47,33,0,0,44,14,0,0,0,0,47,33,0,0,44,14],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[22,8,0,0,31,14,0,0,0,0,39,53,0,0,30,47],[0,0,17,17,0,0,1,44,22,6,0,0,31,39,0,0] >;

C3×C23⋊Dic5 in GAP, Magma, Sage, TeX

C_3\times C_2^3\rtimes {\rm Dic}_5
% in TeX

G:=Group("C3xC2^3:Dic5");
// GroupNames label

G:=SmallGroup(480,112);
// by ID

G=gap.SmallGroup(480,112);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,850,2524,18822]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^10=1,f^2=e^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f^-1=b*c*d,f*c*f^-1=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations

׿
×
𝔽