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G = C2×C9.5He3order 486 = 2·35

Direct product of C2 and C9.5He3

direct product, metabelian, nilpotent (class 3), monomial, 3-elementary

Aliases: C2×C9.5He3, C18.4He3, He3.3C18, C18.13- 1+2, 3- 1+2.1C18, C27⋊C32C6, (C3×C54)⋊2C3, (C3×C27)⋊9C6, C9.4(C2×He3), C9○He3.3C6, (C2×He3).1C9, C6.9(C32⋊C9), C32.3(C3×C18), (C3×C18).24C32, C9.1(C2×3- 1+2), (C2×3- 1+2).1C9, (C2×C27⋊C3)⋊2C3, (C3×C6).3(C3×C9), (C3×C9).33(C3×C6), C3.9(C2×C32⋊C9), (C2×C9○He3).1C3, SmallGroup(486,79)

Series: Derived Chief Lower central Upper central

C1C32 — C2×C9.5He3
C1C3C9C3×C9C9○He3C9.5He3 — C2×C9.5He3
C1C3C32 — C2×C9.5He3
C1C18C3×C18 — C2×C9.5He3

Generators and relations for C2×C9.5He3
 G = < a,b,c,d,e | a2=b9=c3=d3=1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=b3c, ece-1=b3cd-1, ede-1=b6d >

3C3
9C3
3C6
9C6
3C32
3C9
3C9
3C18
3C18
3C3×C6
33- 1+2
3C3×C9
3C27
3C27
33- 1+2
3C27
3C2×3- 1+2
3C3×C18
3C2×3- 1+2
3C54
3C54
3C54

Smallest permutation representation of C2×C9.5He3
On 162 points
Generators in S162
(1 140)(2 141)(3 142)(4 143)(5 144)(6 145)(7 146)(8 147)(9 148)(10 149)(11 150)(12 151)(13 152)(14 153)(15 154)(16 155)(17 156)(18 157)(19 158)(20 159)(21 160)(22 161)(23 162)(24 136)(25 137)(26 138)(27 139)(28 100)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 82)(38 83)(39 84)(40 85)(41 86)(42 87)(43 88)(44 89)(45 90)(46 91)(47 92)(48 93)(49 94)(50 95)(51 96)(52 97)(53 98)(54 99)(55 120)(56 121)(57 122)(58 123)(59 124)(60 125)(61 126)(62 127)(63 128)(64 129)(65 130)(66 131)(67 132)(68 133)(69 134)(70 135)(71 109)(72 110)(73 111)(74 112)(75 113)(76 114)(77 115)(78 116)(79 117)(80 118)(81 119)
(1 4 7 10 13 16 19 22 25)(2 5 8 11 14 17 20 23 26)(3 6 9 12 15 18 21 24 27)(28 31 34 37 40 43 46 49 52)(29 32 35 38 41 44 47 50 53)(30 33 36 39 42 45 48 51 54)(55 58 61 64 67 70 73 76 79)(56 59 62 65 68 71 74 77 80)(57 60 63 66 69 72 75 78 81)(82 85 88 91 94 97 100 103 106)(83 86 89 92 95 98 101 104 107)(84 87 90 93 96 99 102 105 108)(109 112 115 118 121 124 127 130 133)(110 113 116 119 122 125 128 131 134)(111 114 117 120 123 126 129 132 135)(136 139 142 145 148 151 154 157 160)(137 140 143 146 149 152 155 158 161)(138 141 144 147 150 153 156 159 162)
(1 44 114)(2 36 133)(3 37 116)(4 47 117)(5 39 109)(6 40 119)(7 50 120)(8 42 112)(9 43 122)(10 53 123)(11 45 115)(12 46 125)(13 29 126)(14 48 118)(15 49 128)(16 32 129)(17 51 121)(18 52 131)(19 35 132)(20 54 124)(21 28 134)(22 38 135)(23 30 127)(24 31 110)(25 41 111)(26 33 130)(27 34 113)(55 146 95)(56 156 96)(57 148 88)(58 149 98)(59 159 99)(60 151 91)(61 152 101)(62 162 102)(63 154 94)(64 155 104)(65 138 105)(66 157 97)(67 158 107)(68 141 108)(69 160 100)(70 161 83)(71 144 84)(72 136 103)(73 137 86)(74 147 87)(75 139 106)(76 140 89)(77 150 90)(78 142 82)(79 143 92)(80 153 93)(81 145 85)
(2 20 11)(3 12 21)(5 23 14)(6 15 24)(8 26 17)(9 18 27)(29 47 38)(30 39 48)(32 50 41)(33 42 51)(35 53 44)(36 45 54)(55 64 73)(57 75 66)(58 67 76)(60 78 69)(61 70 79)(63 81 72)(83 101 92)(84 93 102)(86 104 95)(87 96 105)(89 107 98)(90 99 108)(110 128 119)(111 120 129)(113 131 122)(114 123 132)(116 134 125)(117 126 135)(136 145 154)(138 156 147)(139 148 157)(141 159 150)(142 151 160)(144 162 153)
(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)(136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162)

G:=sub<Sym(162)| (1,140)(2,141)(3,142)(4,143)(5,144)(6,145)(7,146)(8,147)(9,148)(10,149)(11,150)(12,151)(13,152)(14,153)(15,154)(16,155)(17,156)(18,157)(19,158)(20,159)(21,160)(22,161)(23,162)(24,136)(25,137)(26,138)(27,139)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,82)(38,83)(39,84)(40,85)(41,86)(42,87)(43,88)(44,89)(45,90)(46,91)(47,92)(48,93)(49,94)(50,95)(51,96)(52,97)(53,98)(54,99)(55,120)(56,121)(57,122)(58,123)(59,124)(60,125)(61,126)(62,127)(63,128)(64,129)(65,130)(66,131)(67,132)(68,133)(69,134)(70,135)(71,109)(72,110)(73,111)(74,112)(75,113)(76,114)(77,115)(78,116)(79,117)(80,118)(81,119), (1,4,7,10,13,16,19,22,25)(2,5,8,11,14,17,20,23,26)(3,6,9,12,15,18,21,24,27)(28,31,34,37,40,43,46,49,52)(29,32,35,38,41,44,47,50,53)(30,33,36,39,42,45,48,51,54)(55,58,61,64,67,70,73,76,79)(56,59,62,65,68,71,74,77,80)(57,60,63,66,69,72,75,78,81)(82,85,88,91,94,97,100,103,106)(83,86,89,92,95,98,101,104,107)(84,87,90,93,96,99,102,105,108)(109,112,115,118,121,124,127,130,133)(110,113,116,119,122,125,128,131,134)(111,114,117,120,123,126,129,132,135)(136,139,142,145,148,151,154,157,160)(137,140,143,146,149,152,155,158,161)(138,141,144,147,150,153,156,159,162), (1,44,114)(2,36,133)(3,37,116)(4,47,117)(5,39,109)(6,40,119)(7,50,120)(8,42,112)(9,43,122)(10,53,123)(11,45,115)(12,46,125)(13,29,126)(14,48,118)(15,49,128)(16,32,129)(17,51,121)(18,52,131)(19,35,132)(20,54,124)(21,28,134)(22,38,135)(23,30,127)(24,31,110)(25,41,111)(26,33,130)(27,34,113)(55,146,95)(56,156,96)(57,148,88)(58,149,98)(59,159,99)(60,151,91)(61,152,101)(62,162,102)(63,154,94)(64,155,104)(65,138,105)(66,157,97)(67,158,107)(68,141,108)(69,160,100)(70,161,83)(71,144,84)(72,136,103)(73,137,86)(74,147,87)(75,139,106)(76,140,89)(77,150,90)(78,142,82)(79,143,92)(80,153,93)(81,145,85), (2,20,11)(3,12,21)(5,23,14)(6,15,24)(8,26,17)(9,18,27)(29,47,38)(30,39,48)(32,50,41)(33,42,51)(35,53,44)(36,45,54)(55,64,73)(57,75,66)(58,67,76)(60,78,69)(61,70,79)(63,81,72)(83,101,92)(84,93,102)(86,104,95)(87,96,105)(89,107,98)(90,99,108)(110,128,119)(111,120,129)(113,131,122)(114,123,132)(116,134,125)(117,126,135)(136,145,154)(138,156,147)(139,148,157)(141,159,150)(142,151,160)(144,162,153), (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)(136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162)>;

G:=Group( (1,140)(2,141)(3,142)(4,143)(5,144)(6,145)(7,146)(8,147)(9,148)(10,149)(11,150)(12,151)(13,152)(14,153)(15,154)(16,155)(17,156)(18,157)(19,158)(20,159)(21,160)(22,161)(23,162)(24,136)(25,137)(26,138)(27,139)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,82)(38,83)(39,84)(40,85)(41,86)(42,87)(43,88)(44,89)(45,90)(46,91)(47,92)(48,93)(49,94)(50,95)(51,96)(52,97)(53,98)(54,99)(55,120)(56,121)(57,122)(58,123)(59,124)(60,125)(61,126)(62,127)(63,128)(64,129)(65,130)(66,131)(67,132)(68,133)(69,134)(70,135)(71,109)(72,110)(73,111)(74,112)(75,113)(76,114)(77,115)(78,116)(79,117)(80,118)(81,119), (1,4,7,10,13,16,19,22,25)(2,5,8,11,14,17,20,23,26)(3,6,9,12,15,18,21,24,27)(28,31,34,37,40,43,46,49,52)(29,32,35,38,41,44,47,50,53)(30,33,36,39,42,45,48,51,54)(55,58,61,64,67,70,73,76,79)(56,59,62,65,68,71,74,77,80)(57,60,63,66,69,72,75,78,81)(82,85,88,91,94,97,100,103,106)(83,86,89,92,95,98,101,104,107)(84,87,90,93,96,99,102,105,108)(109,112,115,118,121,124,127,130,133)(110,113,116,119,122,125,128,131,134)(111,114,117,120,123,126,129,132,135)(136,139,142,145,148,151,154,157,160)(137,140,143,146,149,152,155,158,161)(138,141,144,147,150,153,156,159,162), (1,44,114)(2,36,133)(3,37,116)(4,47,117)(5,39,109)(6,40,119)(7,50,120)(8,42,112)(9,43,122)(10,53,123)(11,45,115)(12,46,125)(13,29,126)(14,48,118)(15,49,128)(16,32,129)(17,51,121)(18,52,131)(19,35,132)(20,54,124)(21,28,134)(22,38,135)(23,30,127)(24,31,110)(25,41,111)(26,33,130)(27,34,113)(55,146,95)(56,156,96)(57,148,88)(58,149,98)(59,159,99)(60,151,91)(61,152,101)(62,162,102)(63,154,94)(64,155,104)(65,138,105)(66,157,97)(67,158,107)(68,141,108)(69,160,100)(70,161,83)(71,144,84)(72,136,103)(73,137,86)(74,147,87)(75,139,106)(76,140,89)(77,150,90)(78,142,82)(79,143,92)(80,153,93)(81,145,85), (2,20,11)(3,12,21)(5,23,14)(6,15,24)(8,26,17)(9,18,27)(29,47,38)(30,39,48)(32,50,41)(33,42,51)(35,53,44)(36,45,54)(55,64,73)(57,75,66)(58,67,76)(60,78,69)(61,70,79)(63,81,72)(83,101,92)(84,93,102)(86,104,95)(87,96,105)(89,107,98)(90,99,108)(110,128,119)(111,120,129)(113,131,122)(114,123,132)(116,134,125)(117,126,135)(136,145,154)(138,156,147)(139,148,157)(141,159,150)(142,151,160)(144,162,153), (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)(136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162) );

G=PermutationGroup([[(1,140),(2,141),(3,142),(4,143),(5,144),(6,145),(7,146),(8,147),(9,148),(10,149),(11,150),(12,151),(13,152),(14,153),(15,154),(16,155),(17,156),(18,157),(19,158),(20,159),(21,160),(22,161),(23,162),(24,136),(25,137),(26,138),(27,139),(28,100),(29,101),(30,102),(31,103),(32,104),(33,105),(34,106),(35,107),(36,108),(37,82),(38,83),(39,84),(40,85),(41,86),(42,87),(43,88),(44,89),(45,90),(46,91),(47,92),(48,93),(49,94),(50,95),(51,96),(52,97),(53,98),(54,99),(55,120),(56,121),(57,122),(58,123),(59,124),(60,125),(61,126),(62,127),(63,128),(64,129),(65,130),(66,131),(67,132),(68,133),(69,134),(70,135),(71,109),(72,110),(73,111),(74,112),(75,113),(76,114),(77,115),(78,116),(79,117),(80,118),(81,119)], [(1,4,7,10,13,16,19,22,25),(2,5,8,11,14,17,20,23,26),(3,6,9,12,15,18,21,24,27),(28,31,34,37,40,43,46,49,52),(29,32,35,38,41,44,47,50,53),(30,33,36,39,42,45,48,51,54),(55,58,61,64,67,70,73,76,79),(56,59,62,65,68,71,74,77,80),(57,60,63,66,69,72,75,78,81),(82,85,88,91,94,97,100,103,106),(83,86,89,92,95,98,101,104,107),(84,87,90,93,96,99,102,105,108),(109,112,115,118,121,124,127,130,133),(110,113,116,119,122,125,128,131,134),(111,114,117,120,123,126,129,132,135),(136,139,142,145,148,151,154,157,160),(137,140,143,146,149,152,155,158,161),(138,141,144,147,150,153,156,159,162)], [(1,44,114),(2,36,133),(3,37,116),(4,47,117),(5,39,109),(6,40,119),(7,50,120),(8,42,112),(9,43,122),(10,53,123),(11,45,115),(12,46,125),(13,29,126),(14,48,118),(15,49,128),(16,32,129),(17,51,121),(18,52,131),(19,35,132),(20,54,124),(21,28,134),(22,38,135),(23,30,127),(24,31,110),(25,41,111),(26,33,130),(27,34,113),(55,146,95),(56,156,96),(57,148,88),(58,149,98),(59,159,99),(60,151,91),(61,152,101),(62,162,102),(63,154,94),(64,155,104),(65,138,105),(66,157,97),(67,158,107),(68,141,108),(69,160,100),(70,161,83),(71,144,84),(72,136,103),(73,137,86),(74,147,87),(75,139,106),(76,140,89),(77,150,90),(78,142,82),(79,143,92),(80,153,93),(81,145,85)], [(2,20,11),(3,12,21),(5,23,14),(6,15,24),(8,26,17),(9,18,27),(29,47,38),(30,39,48),(32,50,41),(33,42,51),(35,53,44),(36,45,54),(55,64,73),(57,75,66),(58,67,76),(60,78,69),(61,70,79),(63,81,72),(83,101,92),(84,93,102),(86,104,95),(87,96,105),(89,107,98),(90,99,108),(110,128,119),(111,120,129),(113,131,122),(114,123,132),(116,134,125),(117,126,135),(136,145,154),(138,156,147),(139,148,157),(141,159,150),(142,151,160),(144,162,153)], [(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),(136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162)]])

102 conjugacy classes

class 1  2 3A3B3C3D3E3F6A6B6C6D6E6F9A···9F9G9H9I9J9K9L9M9N18A···18F18G18H18I18J18K18L18M18N27A···27R27S···27AD54A···54R54S···54AD
order123333336666669···99999999918···18181818181818181827···2727···2754···5454···54
size111133991133991···1333399991···1333399993···39···93···39···9

102 irreducible representations

dim111111111111333333
type++
imageC1C2C3C3C3C6C6C6C9C9C18C18He33- 1+2C2×He3C2×3- 1+2C9.5He3C2×C9.5He3
kernelC2×C9.5He3C9.5He3C3×C54C2×C27⋊C3C2×C9○He3C3×C27C27⋊C3C9○He3C2×He3C2×3- 1+2He33- 1+2C18C18C9C9C2C1
# reps1124224261261224241818

Matrix representation of C2×C9.5He3 in GL4(𝔽109) generated by

108000
0100
0010
0001
,
1000
03800
00380
00038
,
1000
0010
0001
0100
,
1000
0100
00450
00063
,
1000
00220
00022
07800
G:=sub<GL(4,GF(109))| [108,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,38,0,0,0,0,38,0,0,0,0,38],[1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,45,0,0,0,0,63],[1,0,0,0,0,0,0,78,0,22,0,0,0,0,22,0] >;

C2×C9.5He3 in GAP, Magma, Sage, TeX

C_2\times C_9._5{\rm He}_3
% in TeX

G:=Group("C2xC9.5He3");
// GroupNames label

G:=SmallGroup(486,79);
// by ID

G=gap.SmallGroup(486,79);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,224,2169,735,118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^9=c^3=d^3=1,e^3=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=b^3*c,e*c*e^-1=b^3*c*d^-1,e*d*e^-1=b^6*d>;
// generators/relations

Export

Subgroup lattice of C2×C9.5He3 in TeX

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