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G = C23×C18order 144 = 24·32

Abelian group of type [2,2,2,18]

direct product, abelian, monomial, 2-elementary

Aliases: C23×C18, SmallGroup(144,113)

Series: Derived Chief Lower central Upper central

C1 — C23×C18
C1C3C9C18C2×C18C22×C18 — C23×C18
C1 — C23×C18
C1 — C23×C18

Generators and relations for C23×C18
 G = < a,b,c,d | a2=b2=c2=d18=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 201, all normal (6 characteristic)
C1, C2 [×15], C3, C22 [×35], C6 [×15], C23 [×15], C9, C2×C6 [×35], C24, C18 [×15], C22×C6 [×15], C2×C18 [×35], C23×C6, C22×C18 [×15], C23×C18
Quotients: C1, C2 [×15], C3, C22 [×35], C6 [×15], C23 [×15], C9, C2×C6 [×35], C24, C18 [×15], C22×C6 [×15], C2×C18 [×35], C23×C6, C22×C18 [×15], C23×C18

Smallest permutation representation of C23×C18
Regular action on 144 points
Generators in S144
(1 81)(2 82)(3 83)(4 84)(5 85)(6 86)(7 87)(8 88)(9 89)(10 90)(11 73)(12 74)(13 75)(14 76)(15 77)(16 78)(17 79)(18 80)(19 93)(20 94)(21 95)(22 96)(23 97)(24 98)(25 99)(26 100)(27 101)(28 102)(29 103)(30 104)(31 105)(32 106)(33 107)(34 108)(35 91)(36 92)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 121)(50 122)(51 123)(52 124)(53 125)(54 126)(55 132)(56 133)(57 134)(58 135)(59 136)(60 137)(61 138)(62 139)(63 140)(64 141)(65 142)(66 143)(67 144)(68 127)(69 128)(70 129)(71 130)(72 131)
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 37)(12 38)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 65)(20 66)(21 67)(22 68)(23 69)(24 70)(25 71)(26 72)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(36 64)(73 109)(74 110)(75 111)(76 112)(77 113)(78 114)(79 115)(80 116)(81 117)(82 118)(83 119)(84 120)(85 121)(86 122)(87 123)(88 124)(89 125)(90 126)(91 140)(92 141)(93 142)(94 143)(95 144)(96 127)(97 128)(98 129)(99 130)(100 131)(101 132)(102 133)(103 134)(104 135)(105 136)(106 137)(107 138)(108 139)
(1 35)(2 36)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(109 132)(110 133)(111 134)(112 135)(113 136)(114 137)(115 138)(116 139)(117 140)(118 141)(119 142)(120 143)(121 144)(122 127)(123 128)(124 129)(125 130)(126 131)
(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)

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

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

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

C23×C18 is a maximal subgroup of   C244D9  C24⋊C27  C2443- 1+2

144 conjugacy classes

class 1 2A···2O3A3B6A···6AD9A···9F18A···18CL
order12···2336···69···918···18
size11···1111···11···11···1

144 irreducible representations

dim111111
type++
imageC1C2C3C6C9C18
kernelC23×C18C22×C18C23×C6C22×C6C24C23
# reps115230690

Matrix representation of C23×C18 in GL4(𝔽19) generated by

1000
0100
00180
0001
,
18000
01800
0010
0001
,
18000
0100
0010
00018
,
8000
0800
0080
0003
G:=sub<GL(4,GF(19))| [1,0,0,0,0,1,0,0,0,0,18,0,0,0,0,1],[18,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[18,0,0,0,0,1,0,0,0,0,1,0,0,0,0,18],[8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,3] >;

C23×C18 in GAP, Magma, Sage, TeX

C_2^3\times C_{18}
% in TeX

G:=Group("C2^3xC18");
// GroupNames label

G:=SmallGroup(144,113);
// by ID

G=gap.SmallGroup(144,113);
# by ID

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

G:=Group<a,b,c,d|a^2=b^2=c^2=d^18=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations

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