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

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

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

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

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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