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G = C3×D44order 264 = 23·3·11

Direct product of C3 and D44

direct product, metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C3×D44, C335D4, C441C6, C1323C2, D221C6, C123D11, C6.15D22, C66.15C22, C4⋊(C3×D11), C111(C3×D4), (C6×D11)⋊4C2, C22.3(C2×C6), C2.4(C6×D11), SmallGroup(264,15)

Series: Derived Chief Lower central Upper central

C1C22 — C3×D44
C1C11C22C66C6×D11 — C3×D44
C11C22 — C3×D44
C1C6C12

Generators and relations for C3×D44
 G = < a,b,c | a3=b44=c2=1, ab=ba, ac=ca, cbc=b-1 >

22C2
22C2
11C22
11C22
22C6
22C6
2D11
2D11
11D4
11C2×C6
11C2×C6
2C3×D11
2C3×D11
11C3×D4

Smallest permutation representation of C3×D44
On 132 points
Generators in S132
(1 62 129)(2 63 130)(3 64 131)(4 65 132)(5 66 89)(6 67 90)(7 68 91)(8 69 92)(9 70 93)(10 71 94)(11 72 95)(12 73 96)(13 74 97)(14 75 98)(15 76 99)(16 77 100)(17 78 101)(18 79 102)(19 80 103)(20 81 104)(21 82 105)(22 83 106)(23 84 107)(24 85 108)(25 86 109)(26 87 110)(27 88 111)(28 45 112)(29 46 113)(30 47 114)(31 48 115)(32 49 116)(33 50 117)(34 51 118)(35 52 119)(36 53 120)(37 54 121)(38 55 122)(39 56 123)(40 57 124)(41 58 125)(42 59 126)(43 60 127)(44 61 128)
(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)
(1 44)(2 43)(3 42)(4 41)(5 40)(6 39)(7 38)(8 37)(9 36)(10 35)(11 34)(12 33)(13 32)(14 31)(15 30)(16 29)(17 28)(18 27)(19 26)(20 25)(21 24)(22 23)(45 78)(46 77)(47 76)(48 75)(49 74)(50 73)(51 72)(52 71)(53 70)(54 69)(55 68)(56 67)(57 66)(58 65)(59 64)(60 63)(61 62)(79 88)(80 87)(81 86)(82 85)(83 84)(89 124)(90 123)(91 122)(92 121)(93 120)(94 119)(95 118)(96 117)(97 116)(98 115)(99 114)(100 113)(101 112)(102 111)(103 110)(104 109)(105 108)(106 107)(125 132)(126 131)(127 130)(128 129)

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

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

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

75 conjugacy classes

class 1 2A2B2C3A3B 4 6A6B6C6D6E6F11A···11E12A12B22A···22E33A···33J44A···44J66A···66J132A···132T
order122233466666611···11121222···2233···3344···4466···66132···132
size11222211211222222222···2222···22···22···22···22···2

75 irreducible representations

dim11111122222222
type+++++++
imageC1C2C2C3C6C6D4D11C3×D4D22C3×D11D44C6×D11C3×D44
kernelC3×D44C132C6×D11D44C44D22C33C12C11C6C4C3C2C1
# reps112224152510101020

Matrix representation of C3×D44 in GL2(𝔽43) generated by

360
036
,
518
4041
,
4142
32
G:=sub<GL(2,GF(43))| [36,0,0,36],[5,40,18,41],[41,3,42,2] >;

C3×D44 in GAP, Magma, Sage, TeX

C_3\times D_{44}
% in TeX

G:=Group("C3xD44");
// GroupNames label

G:=SmallGroup(264,15);
// by ID

G=gap.SmallGroup(264,15);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-11,141,66,6004]);
// Polycyclic

G:=Group<a,b,c|a^3=b^44=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D44 in TeX

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