direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C11⋊D4, C33⋊8D4, D22⋊2C6, Dic11⋊C6, C6.17D22, C66.17C22, C11⋊2(C3×D4), (C2×C22)⋊4C6, (C2×C66)⋊4C2, (C2×C6)⋊1D11, (C6×D11)⋊5C2, C22.5(C2×C6), C2.5(C6×D11), C22⋊2(C3×D11), (C3×Dic11)⋊4C2, SmallGroup(264,17)
Series: Derived ►Chief ►Lower central ►Upper central
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 >
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 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 11A | ··· | 11E | 12A | 12B | 22A | ··· | 22O | 33A | ··· | 33J | 66A | ··· | 66AD |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 11 | ··· | 11 | 12 | 12 | 22 | ··· | 22 | 33 | ··· | 33 | 66 | ··· | 66 |
| size | 1 | 1 | 2 | 22 | 1 | 1 | 22 | 1 | 1 | 2 | 2 | 22 | 22 | 2 | ··· | 2 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
75 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 | C3 | C6 | C6 | C6 | D4 | D11 | C3×D4 | D22 | C3×D11 | C11⋊D4 | C6×D11 | C3×C11⋊D4 |
| kernel | C3×C11⋊D4 | C3×Dic11 | C6×D11 | C2×C66 | C11⋊D4 | Dic11 | D22 | C2×C22 | C33 | C2×C6 | C11 | C6 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 5 | 2 | 5 | 10 | 10 | 10 | 20 |
Matrix representation of C3×C11⋊D4 ►in GL2(𝔽397) generated by
| 34 | 0 |
| 0 | 34 |
| 289 | 396 |
| 1 | 0 |
| 273 | 380 |
| 274 | 124 |
| 1 | 0 |
| 289 | 396 |
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
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