Copied to
clipboard

G = C6×D23order 276 = 22·3·23

Direct product of C6 and D23

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

Aliases: C6×D23, C46⋊C6, C1382C2, C693C22, C23⋊(C2×C6), SmallGroup(276,7)

Series: Derived Chief Lower central Upper central

C1C23 — C6×D23
C1C23C69C3×D23 — C6×D23
C23 — C6×D23
C1C6

Generators and relations for C6×D23
 G = < a,b,c | a6=b23=c2=1, ab=ba, ac=ca, cbc=b-1 >

23C2
23C2
23C22
23C6
23C6
23C2×C6

Smallest permutation representation of C6×D23
On 138 points
Generators in S138
(1 111 56 75 29 116)(2 112 57 76 30 117)(3 113 58 77 31 118)(4 114 59 78 32 119)(5 115 60 79 33 120)(6 93 61 80 34 121)(7 94 62 81 35 122)(8 95 63 82 36 123)(9 96 64 83 37 124)(10 97 65 84 38 125)(11 98 66 85 39 126)(12 99 67 86 40 127)(13 100 68 87 41 128)(14 101 69 88 42 129)(15 102 47 89 43 130)(16 103 48 90 44 131)(17 104 49 91 45 132)(18 105 50 92 46 133)(19 106 51 70 24 134)(20 107 52 71 25 135)(21 108 53 72 26 136)(22 109 54 73 27 137)(23 110 55 74 28 138)
(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)
(1 74)(2 73)(3 72)(4 71)(5 70)(6 92)(7 91)(8 90)(9 89)(10 88)(11 87)(12 86)(13 85)(14 84)(15 83)(16 82)(17 81)(18 80)(19 79)(20 78)(21 77)(22 76)(23 75)(24 115)(25 114)(26 113)(27 112)(28 111)(29 110)(30 109)(31 108)(32 107)(33 106)(34 105)(35 104)(36 103)(37 102)(38 101)(39 100)(40 99)(41 98)(42 97)(43 96)(44 95)(45 94)(46 93)(47 124)(48 123)(49 122)(50 121)(51 120)(52 119)(53 118)(54 117)(55 116)(56 138)(57 137)(58 136)(59 135)(60 134)(61 133)(62 132)(63 131)(64 130)(65 129)(66 128)(67 127)(68 126)(69 125)

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

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

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

78 conjugacy classes

class 1 2A2B2C3A3B6A6B6C6D6E6F23A···23K46A···46K69A···69V138A···138V
order12223366666623···2346···4669···69138···138
size1123231111232323232···22···22···22···2

78 irreducible representations

dim1111112222
type+++++
imageC1C2C2C3C6C6D23D46C3×D23C6×D23
kernelC6×D23C3×D23C138D46D23C46C6C3C2C1
# reps12124211112222

Matrix representation of C6×D23 in GL2(𝔽139) generated by

970
097
,
01
13844
,
0138
1380
G:=sub<GL(2,GF(139))| [97,0,0,97],[0,138,1,44],[0,138,138,0] >;

C6×D23 in GAP, Magma, Sage, TeX

C_6\times D_{23}
% in TeX

G:=Group("C6xD23");
// GroupNames label

G:=SmallGroup(276,7);
// by ID

G=gap.SmallGroup(276,7);
# by ID

G:=PCGroup([4,-2,-2,-3,-23,4227]);
// Polycyclic

G:=Group<a,b,c|a^6=b^23=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C6×D23 in TeX

׿
×
𝔽