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G = C337D4order 264 = 23·3·11

1st semidirect product of C33 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C337D4, D662C2, C2.5D66, C22.12D6, C6.12D22, C222D33, Dic331C2, C66.12C22, (C2×C22)⋊4S3, (C2×C66)⋊2C2, (C2×C6)⋊2D11, C113(C3⋊D4), C33(C11⋊D4), SmallGroup(264,27)

Series: Derived Chief Lower central Upper central

C1C66 — C337D4
C1C11C33C66D66 — C337D4
C33C66 — C337D4
C1C2C22

Generators and relations for C337D4
 G = < a,b,c | a33=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

2C2
66C2
33C22
33C4
2C6
22S3
2C22
6D11
33D4
11D6
11Dic3
3D22
3Dic11
2C66
2D33
11C3⋊D4
3C11⋊D4

Smallest permutation representation of C337D4
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 74 37 128)(2 73 38 127)(3 72 39 126)(4 71 40 125)(5 70 41 124)(6 69 42 123)(7 68 43 122)(8 67 44 121)(9 99 45 120)(10 98 46 119)(11 97 47 118)(12 96 48 117)(13 95 49 116)(14 94 50 115)(15 93 51 114)(16 92 52 113)(17 91 53 112)(18 90 54 111)(19 89 55 110)(20 88 56 109)(21 87 57 108)(22 86 58 107)(23 85 59 106)(24 84 60 105)(25 83 61 104)(26 82 62 103)(27 81 63 102)(28 80 64 101)(29 79 65 100)(30 78 66 132)(31 77 34 131)(32 76 35 130)(33 75 36 129)
(2 33)(3 32)(4 31)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)(17 18)(34 40)(35 39)(36 38)(41 66)(42 65)(43 64)(44 63)(45 62)(46 61)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)(67 102)(68 101)(69 100)(70 132)(71 131)(72 130)(73 129)(74 128)(75 127)(76 126)(77 125)(78 124)(79 123)(80 122)(81 121)(82 120)(83 119)(84 118)(85 117)(86 116)(87 115)(88 114)(89 113)(90 112)(91 111)(92 110)(93 109)(94 108)(95 107)(96 106)(97 105)(98 104)(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), (1,74,37,128)(2,73,38,127)(3,72,39,126)(4,71,40,125)(5,70,41,124)(6,69,42,123)(7,68,43,122)(8,67,44,121)(9,99,45,120)(10,98,46,119)(11,97,47,118)(12,96,48,117)(13,95,49,116)(14,94,50,115)(15,93,51,114)(16,92,52,113)(17,91,53,112)(18,90,54,111)(19,89,55,110)(20,88,56,109)(21,87,57,108)(22,86,58,107)(23,85,59,106)(24,84,60,105)(25,83,61,104)(26,82,62,103)(27,81,63,102)(28,80,64,101)(29,79,65,100)(30,78,66,132)(31,77,34,131)(32,76,35,130)(33,75,36,129), (2,33)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(34,40)(35,39)(36,38)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(67,102)(68,101)(69,100)(70,132)(71,131)(72,130)(73,129)(74,128)(75,127)(76,126)(77,125)(78,124)(79,123)(80,122)(81,121)(82,120)(83,119)(84,118)(85,117)(86,116)(87,115)(88,114)(89,113)(90,112)(91,111)(92,110)(93,109)(94,108)(95,107)(96,106)(97,105)(98,104)(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), (1,74,37,128)(2,73,38,127)(3,72,39,126)(4,71,40,125)(5,70,41,124)(6,69,42,123)(7,68,43,122)(8,67,44,121)(9,99,45,120)(10,98,46,119)(11,97,47,118)(12,96,48,117)(13,95,49,116)(14,94,50,115)(15,93,51,114)(16,92,52,113)(17,91,53,112)(18,90,54,111)(19,89,55,110)(20,88,56,109)(21,87,57,108)(22,86,58,107)(23,85,59,106)(24,84,60,105)(25,83,61,104)(26,82,62,103)(27,81,63,102)(28,80,64,101)(29,79,65,100)(30,78,66,132)(31,77,34,131)(32,76,35,130)(33,75,36,129), (2,33)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(34,40)(35,39)(36,38)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(67,102)(68,101)(69,100)(70,132)(71,131)(72,130)(73,129)(74,128)(75,127)(76,126)(77,125)(78,124)(79,123)(80,122)(81,121)(82,120)(83,119)(84,118)(85,117)(86,116)(87,115)(88,114)(89,113)(90,112)(91,111)(92,110)(93,109)(94,108)(95,107)(96,106)(97,105)(98,104)(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)], [(1,74,37,128),(2,73,38,127),(3,72,39,126),(4,71,40,125),(5,70,41,124),(6,69,42,123),(7,68,43,122),(8,67,44,121),(9,99,45,120),(10,98,46,119),(11,97,47,118),(12,96,48,117),(13,95,49,116),(14,94,50,115),(15,93,51,114),(16,92,52,113),(17,91,53,112),(18,90,54,111),(19,89,55,110),(20,88,56,109),(21,87,57,108),(22,86,58,107),(23,85,59,106),(24,84,60,105),(25,83,61,104),(26,82,62,103),(27,81,63,102),(28,80,64,101),(29,79,65,100),(30,78,66,132),(31,77,34,131),(32,76,35,130),(33,75,36,129)], [(2,33),(3,32),(4,31),(5,30),(6,29),(7,28),(8,27),(9,26),(10,25),(11,24),(12,23),(13,22),(14,21),(15,20),(16,19),(17,18),(34,40),(35,39),(36,38),(41,66),(42,65),(43,64),(44,63),(45,62),(46,61),(47,60),(48,59),(49,58),(50,57),(51,56),(52,55),(53,54),(67,102),(68,101),(69,100),(70,132),(71,131),(72,130),(73,129),(74,128),(75,127),(76,126),(77,125),(78,124),(79,123),(80,122),(81,121),(82,120),(83,119),(84,118),(85,117),(86,116),(87,115),(88,114),(89,113),(90,112),(91,111),(92,110),(93,109),(94,108),(95,107),(96,106),(97,105),(98,104),(99,103)])

69 conjugacy classes

class 1 2A2B2C 3  4 6A6B6C11A···11E22A···22O33A···33J66A···66AD
order12223466611···1122···2233···3366···66
size112662662222···22···22···22···2

69 irreducible representations

dim11112222222222
type+++++++++++
imageC1C2C2C2S3D4D6D11C3⋊D4D22D33C11⋊D4D66C337D4
kernelC337D4Dic33D66C2×C66C2×C22C33C22C2×C6C11C6C22C3C2C1
# reps111111152510101020

Matrix representation of C337D4 in GL2(𝔽397) generated by

1540
361354
,
96265
139301
,
153217
33244
G:=sub<GL(2,GF(397))| [15,361,40,354],[96,139,265,301],[153,33,217,244] >;

C337D4 in GAP, Magma, Sage, TeX

C_{33}\rtimes_7D_4
% in TeX

G:=Group("C33:7D4");
// GroupNames label

G:=SmallGroup(264,27);
// by ID

G=gap.SmallGroup(264,27);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-11,61,323,6004]);
// Polycyclic

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

Export

Subgroup lattice of C337D4 in TeX

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