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## G = C5×C33⋊C2order 270 = 2·33·5

### Direct product of C5 and C33⋊C2

Aliases: C5×C33⋊C2, C333C10, (C3×C15)⋊9S3, C153(C3⋊S3), C324(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 ··· 3M 5A 5B 5C 5D 10A 10B 10C 10D 15A ··· 15AZ order 1 2 3 ··· 3 5 5 5 5 10 10 10 10 15 ··· 15 size 1 27 2 ··· 2 1 1 1 1 27 27 27 27 2 ··· 2

75 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C5 C10 S3 C5×S3 kernel C5×C33⋊C2 C32×C15 C33⋊C2 C33 C3×C15 C32 # reps 1 1 4 4 13 52

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

 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 30 0 0 0 0 1 0 0 0 0 0 0 0 0 30 0 0 0 0 1 30
,
 0 1 0 0 0 0 30 30 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 30 1 0 0 0 0 30 0
,
 0 1 0 0 0 0 30 30 0 0 0 0 0 0 0 1 0 0 0 0 30 30 0 0 0 0 0 0 30 1 0 0 0 0 30 0
,
 30 30 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 30 0 0 0 0 0 0 1 0 0 0 0 0 1 30

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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