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G = C3×C11⋊D4order 264 = 23·3·11

Direct product of C3 and C11⋊D4

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

Aliases: C3×C11⋊D4, C338D4, D222C6, Dic11⋊C6, C6.17D22, C66.17C22, C112(C3×D4), (C2×C22)⋊4C6, (C2×C66)⋊4C2, (C2×C6)⋊1D11, (C6×D11)⋊5C2, C22.5(C2×C6), C2.5(C6×D11), C222(C3×D11), (C3×Dic11)⋊4C2, SmallGroup(264,17)

Series: Derived Chief Lower central Upper central

C1C22 — C3×C11⋊D4
C1C11C22C66C6×D11 — C3×C11⋊D4
C11C22 — C3×C11⋊D4
C1C6C2×C6

Generators and relations for C3×C11⋊D4
 G = < a,b,c,d | a3=b11=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

2C2
22C2
11C22
11C4
2C6
22C6
2D11
2C22
11D4
11C2×C6
11C12
2C66
2C3×D11
11C3×D4

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

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

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

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

75 conjugacy classes

class 1 2A2B2C3A3B 4 6A6B6C6D6E6F11A···11E12A12B22A···22O33A···33J66A···66AD
order122233466666611···11121222···2233···3366···66
size112221122112222222···222222···22···22···2

75 irreducible representations

dim1111111122222222
type+++++++
imageC1C2C2C2C3C6C6C6D4D11C3×D4D22C3×D11C11⋊D4C6×D11C3×C11⋊D4
kernelC3×C11⋊D4C3×Dic11C6×D11C2×C66C11⋊D4Dic11D22C2×C22C33C2×C6C11C6C22C3C2C1
# reps11112222152510101020

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

340
034
,
289396
10
,
273380
274124
,
10
289396
G:=sub<GL(2,GF(397))| [34,0,0,34],[289,1,396,0],[273,274,380,124],[1,289,0,396] >;

C3×C11⋊D4 in GAP, Magma, Sage, TeX

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

G:=Group("C3xC11:D4");
// GroupNames label

G:=SmallGroup(264,17);
// by ID

G=gap.SmallGroup(264,17);
# by ID

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

G:=Group<a,b,c,d|a^3=b^11=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 C3×C11⋊D4 in TeX

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