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G = C11×C3⋊D4order 264 = 23·3·11

Direct product of C11 and C3⋊D4

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

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

C1C6 — C11×C3⋊D4
C1C3C6C66S3×C22 — C11×C3⋊D4
C3C6 — C11×C3⋊D4
C1C22C2×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 >

2C2
6C2
3C22
3C4
2C6
2S3
2C22
6C22
3D4
3C2×C22
3C44
2C66
2S3×C11
3D4×C11

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 2A2B2C 3  4 6A6B6C11A···11J22A···22J22K···22T22U···22AD33A···33J44A···44J66A···66AD
order12223466611···1122···2222···2222···2233···3344···4466···66
size1126262221···11···12···26···62···26···62···2

99 irreducible representations

dim1111111122222222
type+++++++
imageC1C2C2C2C11C22C22C22S3D4D6C3⋊D4S3×C11D4×C11S3×C22C11×C3⋊D4
kernelC11×C3⋊D4C11×Dic3S3×C22C2×C66C3⋊D4Dic3D6C2×C6C2×C22C33C22C11C22C3C2C1
# reps111110101010111210101020

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

2730
0273
,
0396
1396
,
35123
37446
,
01
10
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

Export

Subgroup lattice of C11×C3⋊D4 in TeX

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