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G = C4×C36order 144 = 24·32

Abelian group of type [4,36]

direct product, abelian, monomial, 2-elementary

Aliases: C4×C36, SmallGroup(144,20)

Series: Derived Chief Lower central Upper central

C1 — C4×C36
C1C3C6C2×C6C2×C18C2×C36 — C4×C36
C1 — C4×C36
C1 — C4×C36

Generators and relations for C4×C36
 G = < a,b | a4=b36=1, ab=ba >


Smallest permutation representation of C4×C36
Regular action on 144 points
Generators in S144
(1 81 48 137)(2 82 49 138)(3 83 50 139)(4 84 51 140)(5 85 52 141)(6 86 53 142)(7 87 54 143)(8 88 55 144)(9 89 56 109)(10 90 57 110)(11 91 58 111)(12 92 59 112)(13 93 60 113)(14 94 61 114)(15 95 62 115)(16 96 63 116)(17 97 64 117)(18 98 65 118)(19 99 66 119)(20 100 67 120)(21 101 68 121)(22 102 69 122)(23 103 70 123)(24 104 71 124)(25 105 72 125)(26 106 37 126)(27 107 38 127)(28 108 39 128)(29 73 40 129)(30 74 41 130)(31 75 42 131)(32 76 43 132)(33 77 44 133)(34 78 45 134)(35 79 46 135)(36 80 47 136)
(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)| (1,81,48,137)(2,82,49,138)(3,83,50,139)(4,84,51,140)(5,85,52,141)(6,86,53,142)(7,87,54,143)(8,88,55,144)(9,89,56,109)(10,90,57,110)(11,91,58,111)(12,92,59,112)(13,93,60,113)(14,94,61,114)(15,95,62,115)(16,96,63,116)(17,97,64,117)(18,98,65,118)(19,99,66,119)(20,100,67,120)(21,101,68,121)(22,102,69,122)(23,103,70,123)(24,104,71,124)(25,105,72,125)(26,106,37,126)(27,107,38,127)(28,108,39,128)(29,73,40,129)(30,74,41,130)(31,75,42,131)(32,76,43,132)(33,77,44,133)(34,78,45,134)(35,79,46,135)(36,80,47,136), (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( (1,81,48,137)(2,82,49,138)(3,83,50,139)(4,84,51,140)(5,85,52,141)(6,86,53,142)(7,87,54,143)(8,88,55,144)(9,89,56,109)(10,90,57,110)(11,91,58,111)(12,92,59,112)(13,93,60,113)(14,94,61,114)(15,95,62,115)(16,96,63,116)(17,97,64,117)(18,98,65,118)(19,99,66,119)(20,100,67,120)(21,101,68,121)(22,102,69,122)(23,103,70,123)(24,104,71,124)(25,105,72,125)(26,106,37,126)(27,107,38,127)(28,108,39,128)(29,73,40,129)(30,74,41,130)(31,75,42,131)(32,76,43,132)(33,77,44,133)(34,78,45,134)(35,79,46,135)(36,80,47,136), (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([(1,81,48,137),(2,82,49,138),(3,83,50,139),(4,84,51,140),(5,85,52,141),(6,86,53,142),(7,87,54,143),(8,88,55,144),(9,89,56,109),(10,90,57,110),(11,91,58,111),(12,92,59,112),(13,93,60,113),(14,94,61,114),(15,95,62,115),(16,96,63,116),(17,97,64,117),(18,98,65,118),(19,99,66,119),(20,100,67,120),(21,101,68,121),(22,102,69,122),(23,103,70,123),(24,104,71,124),(25,105,72,125),(26,106,37,126),(27,107,38,127),(28,108,39,128),(29,73,40,129),(30,74,41,130),(31,75,42,131),(32,76,43,132),(33,77,44,133),(34,78,45,134),(35,79,46,135),(36,80,47,136)], [(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)])

C4×C36 is a maximal subgroup of
C42.D9  C36⋊C8  C424D9  C362Q8  C36.6Q8  C422D9  C426D9  C427D9  C423D9  C42⋊C27  C42⋊3- 1+2

144 conjugacy classes

class 1 2A2B2C3A3B4A···4L6A···6F9A···9F12A···12X18A···18R36A···36BT
order1222334···46···69···912···1218···1836···36
size1111111···11···11···11···11···11···1

144 irreducible representations

dim111111111
type++
imageC1C2C3C4C6C9C12C18C36
kernelC4×C36C2×C36C4×C12C36C2×C12C42C12C2×C4C4
# reps1321266241872

Matrix representation of C4×C36 in GL2(𝔽37) generated by

60
01
,
290
02
G:=sub<GL(2,GF(37))| [6,0,0,1],[29,0,0,2] >;

C4×C36 in GAP, Magma, Sage, TeX

C_4\times C_{36}
% in TeX

G:=Group("C4xC36");
// GroupNames label

G:=SmallGroup(144,20);
// by ID

G=gap.SmallGroup(144,20);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-3,72,151,230]);
// Polycyclic

G:=Group<a,b|a^4=b^36=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C4×C36 in TeX

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