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G = D4×C33order 264 = 23·3·11

Direct product of C33 and D4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: D4×C33, C4⋊C66, C443C6, C123C22, C1327C2, C222C66, C66.23C22, (C2×C6)⋊1C22, (C2×C66)⋊1C2, (C2×C22)⋊3C6, C6.6(C2×C22), C22.6(C2×C6), C2.1(C2×C66), SmallGroup(264,29)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C33
C1C2C22C66C2×C66 — D4×C33
C1C2 — D4×C33
C1C66 — D4×C33

Generators and relations for D4×C33
 G = < a,b,c | a33=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

2C2
2C2
2C6
2C6
2C22
2C22
2C66
2C66

Smallest permutation representation of D4×C33
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 49 128 87)(2 50 129 88)(3 51 130 89)(4 52 131 90)(5 53 132 91)(6 54 100 92)(7 55 101 93)(8 56 102 94)(9 57 103 95)(10 58 104 96)(11 59 105 97)(12 60 106 98)(13 61 107 99)(14 62 108 67)(15 63 109 68)(16 64 110 69)(17 65 111 70)(18 66 112 71)(19 34 113 72)(20 35 114 73)(21 36 115 74)(22 37 116 75)(23 38 117 76)(24 39 118 77)(25 40 119 78)(26 41 120 79)(27 42 121 80)(28 43 122 81)(29 44 123 82)(30 45 124 83)(31 46 125 84)(32 47 126 85)(33 48 127 86)
(1 87)(2 88)(3 89)(4 90)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(11 97)(12 98)(13 99)(14 67)(15 68)(16 69)(17 70)(18 71)(19 72)(20 73)(21 74)(22 75)(23 76)(24 77)(25 78)(26 79)(27 80)(28 81)(29 82)(30 83)(31 84)(32 85)(33 86)(34 113)(35 114)(36 115)(37 116)(38 117)(39 118)(40 119)(41 120)(42 121)(43 122)(44 123)(45 124)(46 125)(47 126)(48 127)(49 128)(50 129)(51 130)(52 131)(53 132)(54 100)(55 101)(56 102)(57 103)(58 104)(59 105)(60 106)(61 107)(62 108)(63 109)(64 110)(65 111)(66 112)

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

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

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

165 conjugacy classes

class 1 2A2B2C3A3B 4 6A6B6C6D6E6F11A···11J12A12B22A···22J22K···22AD33A···33T44A···44J66A···66T66U···66BH132A···132T
order122233466666611···11121222···2222···2233···3344···4466···6666···66132···132
size11221121122221···1221···12···21···12···21···12···22···2

165 irreducible representations

dim1111111111112222
type++++
imageC1C2C2C3C6C6C11C22C22C33C66C66D4C3×D4D4×C11D4×C33
kernelD4×C33C132C2×C66D4×C11C44C2×C22C3×D4C12C2×C6D4C4C22C33C11C3C1
# reps112224101020202040121020

Matrix representation of D4×C33 in GL2(𝔽397) generated by

1470
0147
,
01
3960
,
01
10
G:=sub<GL(2,GF(397))| [147,0,0,147],[0,396,1,0],[0,1,1,0] >;

D4×C33 in GAP, Magma, Sage, TeX

D_4\times C_{33}
% in TeX

G:=Group("D4xC33");
// GroupNames label

G:=SmallGroup(264,29);
// by ID

G=gap.SmallGroup(264,29);
# by ID

G:=PCGroup([5,-2,-2,-3,-11,-2,1341]);
// Polycyclic

G:=Group<a,b,c|a^33=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D4×C33 in TeX

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