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## G = C11×C3⋊D4order 264 = 23·3·11

### Direct product of C11 and C3⋊D4

Aliases: C11×C3⋊D4, C339D4, D62C22, Dic3⋊C22, C22.17D6, C66.22C22, C32(D4×C11), (C2×C22)⋊3S3, (C2×C66)⋊6C2, (C2×C6)⋊2C22, (S3×C22)⋊5C2, C2.5(S3×C22), C6.5(C2×C22), C222(S3×C11), (C11×Dic3)⋊4C2, SmallGroup(264,22)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C11×C3⋊D4
 Chief series C1 — C3 — C6 — C66 — S3×C22 — C11×C3⋊D4
 Lower central C3 — C6 — C11×C3⋊D4
 Upper central C1 — C22 — C2×C22

Generators and relations for C11×C3⋊D4
G = < a,b,c,d | a11=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Smallest permutation representation of C11×C3⋊D4
On 132 points
Generators in S132
(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)
(1 116 77)(2 117 67)(3 118 68)(4 119 69)(5 120 70)(6 121 71)(7 111 72)(8 112 73)(9 113 74)(10 114 75)(11 115 76)(12 123 61)(13 124 62)(14 125 63)(15 126 64)(16 127 65)(17 128 66)(18 129 56)(19 130 57)(20 131 58)(21 132 59)(22 122 60)(23 44 54)(24 34 55)(25 35 45)(26 36 46)(27 37 47)(28 38 48)(29 39 49)(30 40 50)(31 41 51)(32 42 52)(33 43 53)(78 92 103)(79 93 104)(80 94 105)(81 95 106)(82 96 107)(83 97 108)(84 98 109)(85 99 110)(86 89 100)(87 90 101)(88 91 102)
(1 29 106 22)(2 30 107 12)(3 31 108 13)(4 32 109 14)(5 33 110 15)(6 23 100 16)(7 24 101 17)(8 25 102 18)(9 26 103 19)(10 27 104 20)(11 28 105 21)(34 90 128 72)(35 91 129 73)(36 92 130 74)(37 93 131 75)(38 94 132 76)(39 95 122 77)(40 96 123 67)(41 97 124 68)(42 98 125 69)(43 99 126 70)(44 89 127 71)(45 88 56 112)(46 78 57 113)(47 79 58 114)(48 80 59 115)(49 81 60 116)(50 82 61 117)(51 83 62 118)(52 84 63 119)(53 85 64 120)(54 86 65 121)(55 87 66 111)
(12 30)(13 31)(14 32)(15 33)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(22 29)(34 66)(35 56)(36 57)(37 58)(38 59)(39 60)(40 61)(41 62)(42 63)(43 64)(44 65)(45 129)(46 130)(47 131)(48 132)(49 122)(50 123)(51 124)(52 125)(53 126)(54 127)(55 128)(67 117)(68 118)(69 119)(70 120)(71 121)(72 111)(73 112)(74 113)(75 114)(76 115)(77 116)(78 92)(79 93)(80 94)(81 95)(82 96)(83 97)(84 98)(85 99)(86 89)(87 90)(88 91)

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

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

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

99 conjugacy classes

 class 1 2A 2B 2C 3 4 6A 6B 6C 11A ··· 11J 22A ··· 22J 22K ··· 22T 22U ··· 22AD 33A ··· 33J 44A ··· 44J 66A ··· 66AD order 1 2 2 2 3 4 6 6 6 11 ··· 11 22 ··· 22 22 ··· 22 22 ··· 22 33 ··· 33 44 ··· 44 66 ··· 66 size 1 1 2 6 2 6 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 6 ··· 6 2 ··· 2 6 ··· 6 2 ··· 2

99 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C11 C22 C22 C22 S3 D4 D6 C3⋊D4 S3×C11 D4×C11 S3×C22 C11×C3⋊D4 kernel C11×C3⋊D4 C11×Dic3 S3×C22 C2×C66 C3⋊D4 Dic3 D6 C2×C6 C2×C22 C33 C22 C11 C22 C3 C2 C1 # reps 1 1 1 1 10 10 10 10 1 1 1 2 10 10 10 20

Matrix representation of C11×C3⋊D4 in GL2(𝔽397) generated by

 273 0 0 273
,
 0 396 1 396
,
 351 23 374 46
,
 0 1 1 0
G:=sub<GL(2,GF(397))| [273,0,0,273],[0,1,396,396],[351,374,23,46],[0,1,1,0] >;

C11×C3⋊D4 in GAP, Magma, Sage, TeX

C_{11}\times C_3\rtimes D_4
% in TeX

G:=Group("C11xC3:D4");
// GroupNames label

G:=SmallGroup(264,22);
// by ID

G=gap.SmallGroup(264,22);
# by ID

G:=PCGroup([5,-2,-2,-11,-2,-3,461,4404]);
// Polycyclic

G:=Group<a,b,c,d|a^11=b^3=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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