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## G = C3×C23⋊Dic5order 480 = 25·3·5

### Direct product of C3 and C23⋊Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C3×C23⋊Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×C30 — C3×C23.D5 — C3×C23⋊Dic5
 Lower central C5 — C10 — C2×C10 — C3×C23⋊Dic5
 Upper central C1 — C6 — C22×C6 — C6×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, C3, C4, C22, C22, C22, C5, C6, C6, C2×C4, C2×C4, D4, C23, C10, C10, C12, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, Dic5, C20, C2×C10, C2×C10, C2×C10, C2×C12, C2×C12, C3×D4, C22×C6, C30, C30, C23⋊C4, C2×Dic5, C2×C20, C5×D4, C22×C10, C3×C22⋊C4, C6×D4, C3×Dic5, C60, C2×C30, C2×C30, C2×C30, C23.D5, D4×C10, C3×C23⋊C4, C6×Dic5, C2×C60, D4×C15, C22×C30, C23⋊Dic5, C3×C23.D5, D4×C30, C3×C23⋊Dic5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, D5, C12, C2×C6, C22⋊C4, Dic5, D10, C2×C12, C3×D4, C3×D5, C23⋊C4, C2×Dic5, C5⋊D4, C3×C22⋊C4, C3×Dic5, C6×D5, C23.D5, C3×C23⋊C4, C6×Dic5, C3×C5⋊D4, C23⋊Dic5, C3×C23.D5, C3×C23⋊Dic5

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

93 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + - - + + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 D4 D5 Dic5 Dic5 D10 C3×D4 C3×D5 C5⋊D4 C3×Dic5 C3×Dic5 C6×D5 C3×C5⋊D4 C23⋊C4 C3×C23⋊C4 C23⋊Dic5 C3×C23⋊Dic5 kernel C3×C23⋊Dic5 C3×C23.D5 D4×C30 C23⋊Dic5 C2×C60 C22×C30 C23.D5 D4×C10 C2×C20 C22×C10 C2×C30 C6×D4 C2×C12 C22×C6 C22×C6 C2×C10 C2×D4 C2×C6 C2×C4 C23 C23 C22 C15 C5 C3 C1 # reps 1 2 1 2 2 2 4 2 4 4 2 2 2 2 2 4 4 8 4 4 4 16 1 2 4 8

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

 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 44 0 0 33 14 0 0 0 0 47 44 0 0 33 14
,
 60 0 0 0 0 60 0 0 0 0 60 0 0 0 0 60
,
 22 31 0 0 8 14 0 0 0 0 39 30 0 0 53 47
,
 0 0 22 31 0 0 6 39 17 1 0 0 17 44 0 0
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