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## G = C2×C6×D11order 264 = 23·3·11

### Direct product of C2×C6 and D11

Aliases: C2×C6×D11, C333C23, C663C22, C22⋊(C2×C6), C11⋊(C22×C6), (C2×C22)⋊5C6, (C2×C66)⋊5C2, SmallGroup(264,36)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C11 — C2×C6×D11
 Chief series C1 — C11 — C33 — C3×D11 — C6×D11 — C2×C6×D11
 Lower central C11 — C2×C6×D11
 Upper central C1 — C2×C6

Generators and relations for C2×C6×D11
G = < a,b,c,d | a2=b6=c11=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 284 in 64 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C3, C22, C22, C6, C6, C23, C11, C2×C6, C2×C6, D11, C22, C22×C6, C33, D22, C2×C22, C3×D11, C66, C22×D11, C6×D11, C2×C66, C2×C6×D11
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, D11, C22×C6, D22, C3×D11, C22×D11, C6×D11, C2×C6×D11

Smallest permutation representation of C2×C6×D11
On 132 points
Generators in S132
(1 109)(2 110)(3 100)(4 101)(5 102)(6 103)(7 104)(8 105)(9 106)(10 107)(11 108)(12 111)(13 112)(14 113)(15 114)(16 115)(17 116)(18 117)(19 118)(20 119)(21 120)(22 121)(23 122)(24 123)(25 124)(26 125)(27 126)(28 127)(29 128)(30 129)(31 130)(32 131)(33 132)(34 67)(35 68)(36 69)(37 70)(38 71)(39 72)(40 73)(41 74)(42 75)(43 76)(44 77)(45 78)(46 79)(47 80)(48 81)(49 82)(50 83)(51 84)(52 85)(53 86)(54 87)(55 88)(56 89)(57 90)(58 91)(59 92)(60 93)(61 94)(62 95)(63 96)(64 97)(65 98)(66 99)
(1 54 32 43 21 65)(2 55 33 44 22 66)(3 45 23 34 12 56)(4 46 24 35 13 57)(5 47 25 36 14 58)(6 48 26 37 15 59)(7 49 27 38 16 60)(8 50 28 39 17 61)(9 51 29 40 18 62)(10 52 30 41 19 63)(11 53 31 42 20 64)(67 111 89 100 78 122)(68 112 90 101 79 123)(69 113 91 102 80 124)(70 114 92 103 81 125)(71 115 93 104 82 126)(72 116 94 105 83 127)(73 117 95 106 84 128)(74 118 96 107 85 129)(75 119 97 108 86 130)(76 120 98 109 87 131)(77 121 99 110 88 132)
(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 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 44)(11 43)(12 51)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 55)(20 54)(21 53)(22 52)(23 62)(24 61)(25 60)(26 59)(27 58)(28 57)(29 56)(30 66)(31 65)(32 64)(33 63)(67 106)(68 105)(69 104)(70 103)(71 102)(72 101)(73 100)(74 110)(75 109)(76 108)(77 107)(78 117)(79 116)(80 115)(81 114)(82 113)(83 112)(84 111)(85 121)(86 120)(87 119)(88 118)(89 128)(90 127)(91 126)(92 125)(93 124)(94 123)(95 122)(96 132)(97 131)(98 130)(99 129)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 6A ··· 6F 6G ··· 6N 11A ··· 11E 22A ··· 22O 33A ··· 33J 66A ··· 66AD order 1 2 2 2 2 2 2 2 3 3 6 ··· 6 6 ··· 6 11 ··· 11 22 ··· 22 33 ··· 33 66 ··· 66 size 1 1 1 1 11 11 11 11 1 1 1 ··· 1 11 ··· 11 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C3 C6 C6 D11 D22 C3×D11 C6×D11 kernel C2×C6×D11 C6×D11 C2×C66 C22×D11 D22 C2×C22 C2×C6 C6 C22 C2 # reps 1 6 1 2 12 2 5 15 10 30

Matrix representation of C2×C6×D11 in GL3(𝔽67) generated by

 66 0 0 0 1 0 0 0 1
,
 38 0 0 0 38 0 0 0 38
,
 1 0 0 0 43 10 0 66 59
,
 1 0 0 0 47 20 0 37 20
G:=sub<GL(3,GF(67))| [66,0,0,0,1,0,0,0,1],[38,0,0,0,38,0,0,0,38],[1,0,0,0,43,66,0,10,59],[1,0,0,0,47,37,0,20,20] >;

C2×C6×D11 in GAP, Magma, Sage, TeX

C_2\times C_6\times D_{11}
% in TeX

G:=Group("C2xC6xD11");
// GroupNames label

G:=SmallGroup(264,36);
// by ID

G=gap.SmallGroup(264,36);
# by ID

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

G:=Group<a,b,c,d|a^2=b^6=c^11=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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