Copied to
clipboard

G = C3×Dic36order 432 = 24·33

Direct product of C3 and Dic36

direct product, metacyclic, supersoluble, monomial

Aliases: C3×Dic36, C72.5C6, C24.5D9, C6.19D36, C12.71D18, Dic18.3C6, C32.3Dic12, C8.(C3×D9), (C3×C9)⋊5Q16, C94(C3×Q16), C4.8(C6×D9), C24.1(C3×S3), (C3×C72).2C2, C2.3(C3×D36), C6.1(C3×D12), C12.64(S3×C6), C36.31(C2×C6), (C3×C24).14S3, (C3×C6).49D12, C18.17(C3×D4), (C3×C18).24D4, (C3×C12).202D6, C3.1(C3×Dic12), (C3×C36).55C22, (C3×Dic18).2C2, SmallGroup(432,104)

Series: Derived Chief Lower central Upper central

C1C36 — C3×Dic36
C1C3C9C18C36C3×C36C3×Dic18 — C3×Dic36
C9C18C36 — C3×Dic36
C1C6C12C24

Generators and relations for C3×Dic36
 G = < a,b,c | a3=b72=1, c2=b36, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 222 in 62 conjugacy classes, 30 normal (26 characteristic)
C1, C2, C3 [×2], C3, C4, C4 [×2], C6 [×2], C6, C8, Q8 [×2], C9, C9, C32, Dic3 [×2], C12 [×2], C12 [×3], Q16, C18, C18, C3×C6, C24 [×2], C24, Dic6 [×2], C3×Q8 [×2], C3×C9, Dic9 [×2], C36, C36, C3×Dic3 [×2], C3×C12, Dic12, C3×Q16, C3×C18, C72, C72, Dic18 [×2], C3×C24, C3×Dic6 [×2], C3×Dic9 [×2], C3×C36, Dic36, C3×Dic12, C3×C72, C3×Dic18 [×2], C3×Dic36
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D4, D6, C2×C6, Q16, D9, C3×S3, D12, C3×D4, D18, S3×C6, Dic12, C3×Q16, C3×D9, D36, C3×D12, C6×D9, Dic36, C3×Dic12, C3×D36, C3×Dic36

Smallest permutation representation of C3×Dic36
On 144 points
Generators in S144
(1 49 25)(2 50 26)(3 51 27)(4 52 28)(5 53 29)(6 54 30)(7 55 31)(8 56 32)(9 57 33)(10 58 34)(11 59 35)(12 60 36)(13 61 37)(14 62 38)(15 63 39)(16 64 40)(17 65 41)(18 66 42)(19 67 43)(20 68 44)(21 69 45)(22 70 46)(23 71 47)(24 72 48)(73 97 121)(74 98 122)(75 99 123)(76 100 124)(77 101 125)(78 102 126)(79 103 127)(80 104 128)(81 105 129)(82 106 130)(83 107 131)(84 108 132)(85 109 133)(86 110 134)(87 111 135)(88 112 136)(89 113 137)(90 114 138)(91 115 139)(92 116 140)(93 117 141)(94 118 142)(95 119 143)(96 120 144)
(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)
(1 122 37 86)(2 121 38 85)(3 120 39 84)(4 119 40 83)(5 118 41 82)(6 117 42 81)(7 116 43 80)(8 115 44 79)(9 114 45 78)(10 113 46 77)(11 112 47 76)(12 111 48 75)(13 110 49 74)(14 109 50 73)(15 108 51 144)(16 107 52 143)(17 106 53 142)(18 105 54 141)(19 104 55 140)(20 103 56 139)(21 102 57 138)(22 101 58 137)(23 100 59 136)(24 99 60 135)(25 98 61 134)(26 97 62 133)(27 96 63 132)(28 95 64 131)(29 94 65 130)(30 93 66 129)(31 92 67 128)(32 91 68 127)(33 90 69 126)(34 89 70 125)(35 88 71 124)(36 87 72 123)

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

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

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

117 conjugacy classes

class 1  2 3A3B3C3D3E4A4B4C6A6B6C6D6E8A8B9A···9I12A···12H12I12J12K12L18A···18I24A···24P36A···36R72A···72AJ
order123333344466666889···912···121212121218···1824···2436···3672···72
size11112222363611222222···22···2363636362···22···22···22···2

117 irreducible representations

dim11111122222222222222222222
type++++++-+++-+-
imageC1C2C2C3C6C6S3D4D6Q16D9C3×S3C3×D4D12D18S3×C6C3×Q16Dic12C3×D9D36C3×D12C6×D9Dic36C3×Dic12C3×D36C3×Dic36
kernelC3×Dic36C3×C72C3×Dic18Dic36C72Dic18C3×C24C3×C18C3×C12C3×C9C24C24C18C3×C6C12C12C9C32C8C6C6C4C3C3C2C1
# reps11222411123222324466461281224

Matrix representation of C3×Dic36 in GL2(𝔽73) generated by

640
064
,
280
060
,
01
720
G:=sub<GL(2,GF(73))| [64,0,0,64],[28,0,0,60],[0,72,1,0] >;

C3×Dic36 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{36}
% in TeX

G:=Group("C3xDic36");
// GroupNames label

G:=SmallGroup(432,104);
// by ID

G=gap.SmallGroup(432,104);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,260,1011,80,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c|a^3=b^72=1,c^2=b^36,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

׿
×
𝔽