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G = C5×C33⋊C2  order 270 = 2·33·5

Direct product of C5 and C33⋊C2

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C5×C33⋊C2, C33⋊3C10, (C3×C15)⋊9S3, C15⋊3(C3⋊S3), C32⋊4(C5×S3), (C32×C15)⋊7C2, C3⋊(C5×C3⋊S3), SmallGroup(270,28)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C33 — C5×C33⋊C2
Chief series
C1 — C3 — C32 — C33 — C32×C15 — C5×C33⋊C2
Lower central
C33 — C5×C33⋊C2
Upper central
C1 — C5

Generators and relations for C5×C33⋊C2
 G = < a,b,c,d,e | a5=b3=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=b-1, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 424 in 112 conjugacy classes, 58 normal (6 characteristic)
C1, C2, C3, C5, S3, C32, C10, C15, C3⋊S3, C33, C5×S3, C3×C15, C33⋊C2, C5×C3⋊S3, C32×C15, C5×C33⋊C2
Quotients: C1, C2, C5, S3, C10, C3⋊S3, C5×S3, C33⋊C2, C5×C3⋊S3, C5×C33⋊C2

Smallest permutation representation of C5×C33⋊C2
►On 135 points
Generators in S135
(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)(121 122 123 124 125)(126 127 128 129 130)(131 132 133 134 135)

(1 79 9)(2 80 10)(3 76 6)(4 77 7)(5 78 8)(11 121 84)(12 122 85)(13 123 81)(14 124 82)(15 125 83)(16 126 31)(17 127 32)(18 128 33)(19 129 34)(20 130 35)(21 109 36)(22 110 37)(23 106 38)(24 107 39)(25 108 40)(26 88 72)(27 89 73)(28 90 74)(29 86 75)(30 87 71)(41 134 61)(42 135 62)(43 131 63)(44 132 64)(45 133 65)(46 59 66)(47 60 67)(48 56 68)(49 57 69)(50 58 70)(51 113 97)(52 114 98)(53 115 99)(54 111 100)(55 112 96)(91 117 101)(92 118 102)(93 119 103)(94 120 104)(95 116 105)

(1 134 123)(2 135 124)(3 131 125)(4 132 121)(5 133 122)(6 43 15)(7 44 11)(8 45 12)(9 41 13)(10 42 14)(16 51 49)(17 52 50)(18 53 46)(19 54 47)(20 55 48)(21 118 90)(22 119 86)(23 120 87)(24 116 88)(25 117 89)(26 39 95)(27 40 91)(28 36 92)(29 37 93)(30 38 94)(31 97 69)(32 98 70)(33 99 66)(34 100 67)(35 96 68)(56 130 112)(57 126 113)(58 127 114)(59 128 115)(60 129 111)(61 81 79)(62 82 80)(63 83 76)(64 84 77)(65 85 78)(71 106 104)(72 107 105)(73 108 101)(74 109 102)(75 110 103)

(1 89 129)(2 90 130)(3 86 126)(4 87 127)(5 88 128)(6 29 16)(7 30 17)(8 26 18)(9 27 19)(10 28 20)(11 94 50)(12 95 46)(13 91 47)(14 92 48)(15 93 49)(21 112 135)(22 113 131)(23 114 132)(24 115 133)(25 111 134)(31 76 75)(32 77 71)(33 78 72)(34 79 73)(35 80 74)(36 55 42)(37 51 43)(38 52 44)(39 53 45)(40 54 41)(56 124 118)(57 125 119)(58 121 120)(59 122 116)(60 123 117)(61 108 100)(62 109 96)(63 110 97)(64 106 98)(65 107 99)(66 85 105)(67 81 101)(68 82 102)(69 83 103)(70 84 104)

(6 76)(7 77)(8 78)(9 79)(10 80)(11 64)(12 65)(13 61)(14 62)(15 63)(16 75)(17 71)(18 72)(19 73)(20 74)(21 56)(22 57)(23 58)(24 59)(25 60)(26 33)(27 34)(28 35)(29 31)(30 32)(36 68)(37 69)(38 70)(39 66)(40 67)(41 81)(42 82)(43 83)(44 84)(45 85)(46 107)(47 108)(48 109)(49 110)(50 106)(51 103)(52 104)(53 105)(54 101)(55 102)(86 126)(87 127)(88 128)(89 129)(90 130)(91 100)(92 96)(93 97)(94 98)(95 99)(111 117)(112 118)(113 119)(114 120)(115 116)(121 132)(122 133)(123 134)(124 135)(125 131)
 
G:=sub<Sym(135)| (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)(121,122,123,124,125)(126,127,128,129,130)(131,132,133,134,135), (1,79,9)(2,80,10)(3,76,6)(4,77,7)(5,78,8)(11,121,84)(12,122,85)(13,123,81)(14,124,82)(15,125,83)(16,126,31)(17,127,32)(18,128,33)(19,129,34)(20,130,35)(21,109,36)(22,110,37)(23,106,38)(24,107,39)(25,108,40)(26,88,72)(27,89,73)(28,90,74)(29,86,75)(30,87,71)(41,134,61)(42,135,62)(43,131,63)(44,132,64)(45,133,65)(46,59,66)(47,60,67)(48,56,68)(49,57,69)(50,58,70)(51,113,97)(52,114,98)(53,115,99)(54,111,100)(55,112,96)(91,117,101)(92,118,102)(93,119,103)(94,120,104)(95,116,105), (1,134,123)(2,135,124)(3,131,125)(4,132,121)(5,133,122)(6,43,15)(7,44,11)(8,45,12)(9,41,13)(10,42,14)(16,51,49)(17,52,50)(18,53,46)(19,54,47)(20,55,48)(21,118,90)(22,119,86)(23,120,87)(24,116,88)(25,117,89)(26,39,95)(27,40,91)(28,36,92)(29,37,93)(30,38,94)(31,97,69)(32,98,70)(33,99,66)(34,100,67)(35,96,68)(56,130,112)(57,126,113)(58,127,114)(59,128,115)(60,129,111)(61,81,79)(62,82,80)(63,83,76)(64,84,77)(65,85,78)(71,106,104)(72,107,105)(73,108,101)(74,109,102)(75,110,103), (1,89,129)(2,90,130)(3,86,126)(4,87,127)(5,88,128)(6,29,16)(7,30,17)(8,26,18)(9,27,19)(10,28,20)(11,94,50)(12,95,46)(13,91,47)(14,92,48)(15,93,49)(21,112,135)(22,113,131)(23,114,132)(24,115,133)(25,111,134)(31,76,75)(32,77,71)(33,78,72)(34,79,73)(35,80,74)(36,55,42)(37,51,43)(38,52,44)(39,53,45)(40,54,41)(56,124,118)(57,125,119)(58,121,120)(59,122,116)(60,123,117)(61,108,100)(62,109,96)(63,110,97)(64,106,98)(65,107,99)(66,85,105)(67,81,101)(68,82,102)(69,83,103)(70,84,104), (6,76)(7,77)(8,78)(9,79)(10,80)(11,64)(12,65)(13,61)(14,62)(15,63)(16,75)(17,71)(18,72)(19,73)(20,74)(21,56)(22,57)(23,58)(24,59)(25,60)(26,33)(27,34)(28,35)(29,31)(30,32)(36,68)(37,69)(38,70)(39,66)(40,67)(41,81)(42,82)(43,83)(44,84)(45,85)(46,107)(47,108)(48,109)(49,110)(50,106)(51,103)(52,104)(53,105)(54,101)(55,102)(86,126)(87,127)(88,128)(89,129)(90,130)(91,100)(92,96)(93,97)(94,98)(95,99)(111,117)(112,118)(113,119)(114,120)(115,116)(121,132)(122,133)(123,134)(124,135)(125,131)>;
 
G:=Group( (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)(121,122,123,124,125)(126,127,128,129,130)(131,132,133,134,135), (1,79,9)(2,80,10)(3,76,6)(4,77,7)(5,78,8)(11,121,84)(12,122,85)(13,123,81)(14,124,82)(15,125,83)(16,126,31)(17,127,32)(18,128,33)(19,129,34)(20,130,35)(21,109,36)(22,110,37)(23,106,38)(24,107,39)(25,108,40)(26,88,72)(27,89,73)(28,90,74)(29,86,75)(30,87,71)(41,134,61)(42,135,62)(43,131,63)(44,132,64)(45,133,65)(46,59,66)(47,60,67)(48,56,68)(49,57,69)(50,58,70)(51,113,97)(52,114,98)(53,115,99)(54,111,100)(55,112,96)(91,117,101)(92,118,102)(93,119,103)(94,120,104)(95,116,105), (1,134,123)(2,135,124)(3,131,125)(4,132,121)(5,133,122)(6,43,15)(7,44,11)(8,45,12)(9,41,13)(10,42,14)(16,51,49)(17,52,50)(18,53,46)(19,54,47)(20,55,48)(21,118,90)(22,119,86)(23,120,87)(24,116,88)(25,117,89)(26,39,95)(27,40,91)(28,36,92)(29,37,93)(30,38,94)(31,97,69)(32,98,70)(33,99,66)(34,100,67)(35,96,68)(56,130,112)(57,126,113)(58,127,114)(59,128,115)(60,129,111)(61,81,79)(62,82,80)(63,83,76)(64,84,77)(65,85,78)(71,106,104)(72,107,105)(73,108,101)(74,109,102)(75,110,103), (1,89,129)(2,90,130)(3,86,126)(4,87,127)(5,88,128)(6,29,16)(7,30,17)(8,26,18)(9,27,19)(10,28,20)(11,94,50)(12,95,46)(13,91,47)(14,92,48)(15,93,49)(21,112,135)(22,113,131)(23,114,132)(24,115,133)(25,111,134)(31,76,75)(32,77,71)(33,78,72)(34,79,73)(35,80,74)(36,55,42)(37,51,43)(38,52,44)(39,53,45)(40,54,41)(56,124,118)(57,125,119)(58,121,120)(59,122,116)(60,123,117)(61,108,100)(62,109,96)(63,110,97)(64,106,98)(65,107,99)(66,85,105)(67,81,101)(68,82,102)(69,83,103)(70,84,104), (6,76)(7,77)(8,78)(9,79)(10,80)(11,64)(12,65)(13,61)(14,62)(15,63)(16,75)(17,71)(18,72)(19,73)(20,74)(21,56)(22,57)(23,58)(24,59)(25,60)(26,33)(27,34)(28,35)(29,31)(30,32)(36,68)(37,69)(38,70)(39,66)(40,67)(41,81)(42,82)(43,83)(44,84)(45,85)(46,107)(47,108)(48,109)(49,110)(50,106)(51,103)(52,104)(53,105)(54,101)(55,102)(86,126)(87,127)(88,128)(89,129)(90,130)(91,100)(92,96)(93,97)(94,98)(95,99)(111,117)(112,118)(113,119)(114,120)(115,116)(121,132)(122,133)(123,134)(124,135)(125,131) );
 
G=PermutationGroup([[(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),(121,122,123,124,125),(126,127,128,129,130),(131,132,133,134,135)], [(1,79,9),(2,80,10),(3,76,6),(4,77,7),(5,78,8),(11,121,84),(12,122,85),(13,123,81),(14,124,82),(15,125,83),(16,126,31),(17,127,32),(18,128,33),(19,129,34),(20,130,35),(21,109,36),(22,110,37),(23,106,38),(24,107,39),(25,108,40),(26,88,72),(27,89,73),(28,90,74),(29,86,75),(30,87,71),(41,134,61),(42,135,62),(43,131,63),(44,132,64),(45,133,65),(46,59,66),(47,60,67),(48,56,68),(49,57,69),(50,58,70),(51,113,97),(52,114,98),(53,115,99),(54,111,100),(55,112,96),(91,117,101),(92,118,102),(93,119,103),(94,120,104),(95,116,105)], [(1,134,123),(2,135,124),(3,131,125),(4,132,121),(5,133,122),(6,43,15),(7,44,11),(8,45,12),(9,41,13),(10,42,14),(16,51,49),(17,52,50),(18,53,46),(19,54,47),(20,55,48),(21,118,90),(22,119,86),(23,120,87),(24,116,88),(25,117,89),(26,39,95),(27,40,91),(28,36,92),(29,37,93),(30,38,94),(31,97,69),(32,98,70),(33,99,66),(34,100,67),(35,96,68),(56,130,112),(57,126,113),(58,127,114),(59,128,115),(60,129,111),(61,81,79),(62,82,80),(63,83,76),(64,84,77),(65,85,78),(71,106,104),(72,107,105),(73,108,101),(74,109,102),(75,110,103)], [(1,89,129),(2,90,130),(3,86,126),(4,87,127),(5,88,128),(6,29,16),(7,30,17),(8,26,18),(9,27,19),(10,28,20),(11,94,50),(12,95,46),(13,91,47),(14,92,48),(15,93,49),(21,112,135),(22,113,131),(23,114,132),(24,115,133),(25,111,134),(31,76,75),(32,77,71),(33,78,72),(34,79,73),(35,80,74),(36,55,42),(37,51,43),(38,52,44),(39,53,45),(40,54,41),(56,124,118),(57,125,119),(58,121,120),(59,122,116),(60,123,117),(61,108,100),(62,109,96),(63,110,97),(64,106,98),(65,107,99),(66,85,105),(67,81,101),(68,82,102),(69,83,103),(70,84,104)], [(6,76),(7,77),(8,78),(9,79),(10,80),(11,64),(12,65),(13,61),(14,62),(15,63),(16,75),(17,71),(18,72),(19,73),(20,74),(21,56),(22,57),(23,58),(24,59),(25,60),(26,33),(27,34),(28,35),(29,31),(30,32),(36,68),(37,69),(38,70),(39,66),(40,67),(41,81),(42,82),(43,83),(44,84),(45,85),(46,107),(47,108),(48,109),(49,110),(50,106),(51,103),(52,104),(53,105),(54,101),(55,102),(86,126),(87,127),(88,128),(89,129),(90,130),(91,100),(92,96),(93,97),(94,98),(95,99),(111,117),(112,118),(113,119),(114,120),(115,116),(121,132),(122,133),(123,134),(124,135),(125,131)]])
 

75 conjugacy classes

class 1  2 3A···3M5A5B5C5D10A10B10C10D15A···15AZ
order123···355551010101015···15
size1272···21111272727272···2

75 irreducible representations

dim111122
type+++
imageC1C2C5C10S3C5×S3
kernelC5×C33⋊C2C32×C15C33⋊C2C33C3×C15C32
# reps11441352

Matrix representation of C5×C33⋊C2 ►in GL6(𝔽31)

200000
020000
008000
000800
0000160
0000016
,
100000
010000
00303000
001000
0000030
0000130
,
010000
30300000
001000
000100
0000301
0000300
,
010000
30300000
000100
00303000
0000301
0000300
,
30300000
010000
001100
0003000
000010
0000130

G:=sub<GL(6,GF(31))| [2,0,0,0,0,0,0,2,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,1,0,0,0,0,30,0,0,0,0,0,0,0,0,1,0,0,0,0,30,30],[0,30,0,0,0,0,1,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,30,0,0,0,0,1,0],[0,30,0,0,0,0,1,30,0,0,0,0,0,0,0,30,0,0,0,0,1,30,0,0,0,0,0,0,30,30,0,0,0,0,1,0],[30,0,0,0,0,0,30,1,0,0,0,0,0,0,1,0,0,0,0,0,1,30,0,0,0,0,0,0,1,1,0,0,0,0,0,30] >;
 

C5×C33⋊C2 in GAP, Magma, Sage, TeX 

C_5\times C_3^3\rtimes C_2
 
% in TeX
 
G:=Group("C5xC3^3:C2");
 
// GroupNames label
 
G:=SmallGroup(270,28);
 
// by ID
 
G=gap.SmallGroup(270,28);
 
# by ID
 
G:=PCGroup([5,-2,-5,-3,-3,-3,302,1203,4504]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^5=b^3=c^3=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e=b^-1,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
 
// generators/relations
 

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