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

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

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

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

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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