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## G = C33⋊7D4order 264 = 23·3·11

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

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

 Derived series C1 — C66 — C33⋊7D4
 Chief series C1 — C11 — C33 — C66 — D66 — C33⋊7D4
 Lower central C33 — C66 — C33⋊7D4
 Upper central C1 — C2 — C22

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

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

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

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

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

69 conjugacy classes

 class 1 2A 2B 2C 3 4 6A 6B 6C 11A ··· 11E 22A ··· 22O 33A ··· 33J 66A ··· 66AD order 1 2 2 2 3 4 6 6 6 11 ··· 11 22 ··· 22 33 ··· 33 66 ··· 66 size 1 1 2 66 2 66 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

69 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 S3 D4 D6 D11 C3⋊D4 D22 D33 C11⋊D4 D66 C33⋊7D4 kernel C33⋊7D4 Dic33 D66 C2×C66 C2×C22 C33 C22 C2×C6 C11 C6 C22 C3 C2 C1 # reps 1 1 1 1 1 1 1 5 2 5 10 10 10 20

Matrix representation of C337D4 in GL2(𝔽397) generated by

 15 40 361 354
,
 96 265 139 301
,
 153 217 33 244
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

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