Copied to
clipboard

## G = C22×C36order 144 = 24·32

### Abelian group of type [2,2,36]

Aliases: C22×C36, SmallGroup(144,47)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C36
 Chief series C1 — C3 — C6 — C18 — C36 — C2×C36 — C22×C36
 Lower central C1 — C22×C36
 Upper central C1 — C22×C36

Generators and relations for C22×C36
G = < a,b,c | a2=b2=c36=1, ab=ba, ac=ca, bc=cb >

Subgroups: 81, all normal (12 characteristic)
C1, C2, C2 [×6], C3, C4 [×4], C22 [×7], C6, C6 [×6], C2×C4 [×6], C23, C9, C12 [×4], C2×C6 [×7], C22×C4, C18, C18 [×6], C2×C12 [×6], C22×C6, C36 [×4], C2×C18 [×7], C22×C12, C2×C36 [×6], C22×C18, C22×C36
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], C23, C9, C12 [×4], C2×C6 [×7], C22×C4, C18 [×7], C2×C12 [×6], C22×C6, C36 [×4], C2×C18 [×7], C22×C12, C2×C36 [×6], C22×C18, C22×C36

Smallest permutation representation of C22×C36
Regular action on 144 points
Generators in S144
(1 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 71)(10 72)(11 37)(12 38)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 57)(32 58)(33 59)(34 60)(35 61)(36 62)(73 132)(74 133)(75 134)(76 135)(77 136)(78 137)(79 138)(80 139)(81 140)(82 141)(83 142)(84 143)(85 144)(86 109)(87 110)(88 111)(89 112)(90 113)(91 114)(92 115)(93 116)(94 117)(95 118)(96 119)(97 120)(98 121)(99 122)(100 123)(101 124)(102 125)(103 126)(104 127)(105 128)(106 129)(107 130)(108 131)
(1 99)(2 100)(3 101)(4 102)(5 103)(6 104)(7 105)(8 106)(9 107)(10 108)(11 73)(12 74)(13 75)(14 76)(15 77)(16 78)(17 79)(18 80)(19 81)(20 82)(21 83)(22 84)(23 85)(24 86)(25 87)(26 88)(27 89)(28 90)(29 91)(30 92)(31 93)(32 94)(33 95)(34 96)(35 97)(36 98)(37 132)(38 133)(39 134)(40 135)(41 136)(42 137)(43 138)(44 139)(45 140)(46 141)(47 142)(48 143)(49 144)(50 109)(51 110)(52 111)(53 112)(54 113)(55 114)(56 115)(57 116)(58 117)(59 118)(60 119)(61 120)(62 121)(63 122)(64 123)(65 124)(66 125)(67 126)(68 127)(69 128)(70 129)(71 130)(72 131)
(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)| (1,63)(2,64)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,71)(10,72)(11,37)(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,57)(32,58)(33,59)(34,60)(35,61)(36,62)(73,132)(74,133)(75,134)(76,135)(77,136)(78,137)(79,138)(80,139)(81,140)(82,141)(83,142)(84,143)(85,144)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119)(97,120)(98,121)(99,122)(100,123)(101,124)(102,125)(103,126)(104,127)(105,128)(106,129)(107,130)(108,131), (1,99)(2,100)(3,101)(4,102)(5,103)(6,104)(7,105)(8,106)(9,107)(10,108)(11,73)(12,74)(13,75)(14,76)(15,77)(16,78)(17,79)(18,80)(19,81)(20,82)(21,83)(22,84)(23,85)(24,86)(25,87)(26,88)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(33,95)(34,96)(35,97)(36,98)(37,132)(38,133)(39,134)(40,135)(41,136)(42,137)(43,138)(44,139)(45,140)(46,141)(47,142)(48,143)(49,144)(50,109)(51,110)(52,111)(53,112)(54,113)(55,114)(56,115)(57,116)(58,117)(59,118)(60,119)(61,120)(62,121)(63,122)(64,123)(65,124)(66,125)(67,126)(68,127)(69,128)(70,129)(71,130)(72,131), (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( (1,63)(2,64)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,71)(10,72)(11,37)(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,57)(32,58)(33,59)(34,60)(35,61)(36,62)(73,132)(74,133)(75,134)(76,135)(77,136)(78,137)(79,138)(80,139)(81,140)(82,141)(83,142)(84,143)(85,144)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119)(97,120)(98,121)(99,122)(100,123)(101,124)(102,125)(103,126)(104,127)(105,128)(106,129)(107,130)(108,131), (1,99)(2,100)(3,101)(4,102)(5,103)(6,104)(7,105)(8,106)(9,107)(10,108)(11,73)(12,74)(13,75)(14,76)(15,77)(16,78)(17,79)(18,80)(19,81)(20,82)(21,83)(22,84)(23,85)(24,86)(25,87)(26,88)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(33,95)(34,96)(35,97)(36,98)(37,132)(38,133)(39,134)(40,135)(41,136)(42,137)(43,138)(44,139)(45,140)(46,141)(47,142)(48,143)(49,144)(50,109)(51,110)(52,111)(53,112)(54,113)(55,114)(56,115)(57,116)(58,117)(59,118)(60,119)(61,120)(62,121)(63,122)(64,123)(65,124)(66,125)(67,126)(68,127)(69,128)(70,129)(71,130)(72,131), (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([(1,63),(2,64),(3,65),(4,66),(5,67),(6,68),(7,69),(8,70),(9,71),(10,72),(11,37),(12,38),(13,39),(14,40),(15,41),(16,42),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,49),(24,50),(25,51),(26,52),(27,53),(28,54),(29,55),(30,56),(31,57),(32,58),(33,59),(34,60),(35,61),(36,62),(73,132),(74,133),(75,134),(76,135),(77,136),(78,137),(79,138),(80,139),(81,140),(82,141),(83,142),(84,143),(85,144),(86,109),(87,110),(88,111),(89,112),(90,113),(91,114),(92,115),(93,116),(94,117),(95,118),(96,119),(97,120),(98,121),(99,122),(100,123),(101,124),(102,125),(103,126),(104,127),(105,128),(106,129),(107,130),(108,131)], [(1,99),(2,100),(3,101),(4,102),(5,103),(6,104),(7,105),(8,106),(9,107),(10,108),(11,73),(12,74),(13,75),(14,76),(15,77),(16,78),(17,79),(18,80),(19,81),(20,82),(21,83),(22,84),(23,85),(24,86),(25,87),(26,88),(27,89),(28,90),(29,91),(30,92),(31,93),(32,94),(33,95),(34,96),(35,97),(36,98),(37,132),(38,133),(39,134),(40,135),(41,136),(42,137),(43,138),(44,139),(45,140),(46,141),(47,142),(48,143),(49,144),(50,109),(51,110),(52,111),(53,112),(54,113),(55,114),(56,115),(57,116),(58,117),(59,118),(60,119),(61,120),(62,121),(63,122),(64,123),(65,124),(66,125),(67,126),(68,127),(69,128),(70,129),(71,130),(72,131)], [(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)])

C22×C36 is a maximal subgroup of   C36.55D4  C18.C42  C36.49D4  C23.26D18  C23.28D18  C367D4

144 conjugacy classes

 class 1 2A ··· 2G 3A 3B 4A ··· 4H 6A ··· 6N 9A ··· 9F 12A ··· 12P 18A ··· 18AP 36A ··· 36AV order 1 2 ··· 2 3 3 4 ··· 4 6 ··· 6 9 ··· 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 ··· 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C3 C4 C6 C6 C9 C12 C18 C18 C36 kernel C22×C36 C2×C36 C22×C18 C22×C12 C2×C18 C2×C12 C22×C6 C22×C4 C2×C6 C2×C4 C23 C22 # reps 1 6 1 2 8 12 2 6 16 36 6 48

Matrix representation of C22×C36 in GL3(𝔽37) generated by

 36 0 0 0 36 0 0 0 36
,
 36 0 0 0 36 0 0 0 1
,
 12 0 0 0 20 0 0 0 27
G:=sub<GL(3,GF(37))| [36,0,0,0,36,0,0,0,36],[36,0,0,0,36,0,0,0,1],[12,0,0,0,20,0,0,0,27] >;

C22×C36 in GAP, Magma, Sage, TeX

C_2^2\times C_{36}
% in TeX

G:=Group("C2^2xC36");
// GroupNames label

G:=SmallGroup(144,47);
// by ID

G=gap.SmallGroup(144,47);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-3,144,165]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^36=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

׿
×
𝔽