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G = C4×D33order 264 = 23·3·11

Direct product of C4 and D33

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C4×D33, C442S3, C1322C2, C122D11, C6.9D22, C2.1D66, C22.9D6, D66.2C2, Dic335C2, C66.9C22, C112(C4×S3), C334(C2×C4), C32(C4×D11), SmallGroup(264,24)

Series: Derived Chief Lower central Upper central

C1C33 — C4×D33
C1C11C33C66D66 — C4×D33
C33 — C4×D33
C1C4

Generators and relations for C4×D33
 G = < a,b,c | a4=b33=c2=1, ab=ba, ac=ca, cbc=b-1 >

33C2
33C2
33C4
33C22
11S3
11S3
3D11
3D11
33C2×C4
11Dic3
11D6
3D22
3Dic11
11C4×S3
3C4×D11

Smallest permutation representation of C4×D33
On 132 points
Generators in S132
(1 104 62 83)(2 105 63 84)(3 106 64 85)(4 107 65 86)(5 108 66 87)(6 109 34 88)(7 110 35 89)(8 111 36 90)(9 112 37 91)(10 113 38 92)(11 114 39 93)(12 115 40 94)(13 116 41 95)(14 117 42 96)(15 118 43 97)(16 119 44 98)(17 120 45 99)(18 121 46 67)(19 122 47 68)(20 123 48 69)(21 124 49 70)(22 125 50 71)(23 126 51 72)(24 127 52 73)(25 128 53 74)(26 129 54 75)(27 130 55 76)(28 131 56 77)(29 132 57 78)(30 100 58 79)(31 101 59 80)(32 102 60 81)(33 103 61 82)
(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 61)(2 60)(3 59)(4 58)(5 57)(6 56)(7 55)(8 54)(9 53)(10 52)(11 51)(12 50)(13 49)(14 48)(15 47)(16 46)(17 45)(18 44)(19 43)(20 42)(21 41)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 66)(30 65)(31 64)(32 63)(33 62)(67 119)(68 118)(69 117)(70 116)(71 115)(72 114)(73 113)(74 112)(75 111)(76 110)(77 109)(78 108)(79 107)(80 106)(81 105)(82 104)(83 103)(84 102)(85 101)(86 100)(87 132)(88 131)(89 130)(90 129)(91 128)(92 127)(93 126)(94 125)(95 124)(96 123)(97 122)(98 121)(99 120)

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

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

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

72 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D 6 11A···11E12A12B22A···22E33A···33J44A···44J66A···66J132A···132T
order122234444611···11121222···2233···3344···4466···66132···132
size113333211333322···2222···22···22···22···22···2

72 irreducible representations

dim11111222222222
type++++++++++
imageC1C2C2C2C4S3D6D11C4×S3D22D33C4×D11D66C4×D33
kernelC4×D33Dic33C132D66D33C44C22C12C11C6C4C3C2C1
# reps111141152510101020

Matrix representation of C4×D33 in GL4(𝔽397) generated by

63000
06300
003960
000396
,
508200
15324300
00141156
0081374
,
39626600
0100
0023221
00216165
G:=sub<GL(4,GF(397))| [63,0,0,0,0,63,0,0,0,0,396,0,0,0,0,396],[50,153,0,0,82,243,0,0,0,0,141,81,0,0,156,374],[396,0,0,0,266,1,0,0,0,0,232,216,0,0,21,165] >;

C4×D33 in GAP, Magma, Sage, TeX

C_4\times D_{33}
% in TeX

G:=Group("C4xD33");
// GroupNames label

G:=SmallGroup(264,24);
// by ID

G=gap.SmallGroup(264,24);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-11,26,323,6004]);
// Polycyclic

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

Export

Subgroup lattice of C4×D33 in TeX

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