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## G = C3⋊D33order 198 = 2·32·11

### The semidirect product of C3 and D33 acting via D33/C33=C2

Aliases: C3⋊D33, C331S3, C322D11, C11⋊(C3⋊S3), (C3×C33)⋊1C2, SmallGroup(198,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C33 — C3⋊D33
 Chief series C1 — C11 — C33 — C3×C33 — C3⋊D33
 Lower central C3×C33 — C3⋊D33
 Upper central C1

Generators and relations for C3⋊D33
G = < a,b,c | a3=b33=c2=1, ab=ba, cac=a-1, cbc=b-1 >

Smallest permutation representation of C3⋊D33
On 99 points
Generators in S99
(1 92 65)(2 93 66)(3 94 34)(4 95 35)(5 96 36)(6 97 37)(7 98 38)(8 99 39)(9 67 40)(10 68 41)(11 69 42)(12 70 43)(13 71 44)(14 72 45)(15 73 46)(16 74 47)(17 75 48)(18 76 49)(19 77 50)(20 78 51)(21 79 52)(22 80 53)(23 81 54)(24 82 55)(25 83 56)(26 84 57)(27 85 58)(28 86 59)(29 87 60)(30 88 61)(31 89 62)(32 90 63)(33 91 64)
(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)
(1 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(34 89)(35 88)(36 87)(37 86)(38 85)(39 84)(40 83)(41 82)(42 81)(43 80)(44 79)(45 78)(46 77)(47 76)(48 75)(49 74)(50 73)(51 72)(52 71)(53 70)(54 69)(55 68)(56 67)(57 99)(58 98)(59 97)(60 96)(61 95)(62 94)(63 93)(64 92)(65 91)(66 90)

G:=sub<Sym(99)| (1,92,65)(2,93,66)(3,94,34)(4,95,35)(5,96,36)(6,97,37)(7,98,38)(8,99,39)(9,67,40)(10,68,41)(11,69,42)(12,70,43)(13,71,44)(14,72,45)(15,73,46)(16,74,47)(17,75,48)(18,76,49)(19,77,50)(20,78,51)(21,79,52)(22,80,53)(23,81,54)(24,82,55)(25,83,56)(26,84,57)(27,85,58)(28,86,59)(29,87,60)(30,88,61)(31,89,62)(32,90,63)(33,91,64), (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,89)(35,88)(36,87)(37,86)(38,85)(39,84)(40,83)(41,82)(42,81)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,99)(58,98)(59,97)(60,96)(61,95)(62,94)(63,93)(64,92)(65,91)(66,90)>;

G:=Group( (1,92,65)(2,93,66)(3,94,34)(4,95,35)(5,96,36)(6,97,37)(7,98,38)(8,99,39)(9,67,40)(10,68,41)(11,69,42)(12,70,43)(13,71,44)(14,72,45)(15,73,46)(16,74,47)(17,75,48)(18,76,49)(19,77,50)(20,78,51)(21,79,52)(22,80,53)(23,81,54)(24,82,55)(25,83,56)(26,84,57)(27,85,58)(28,86,59)(29,87,60)(30,88,61)(31,89,62)(32,90,63)(33,91,64), (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,89)(35,88)(36,87)(37,86)(38,85)(39,84)(40,83)(41,82)(42,81)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,99)(58,98)(59,97)(60,96)(61,95)(62,94)(63,93)(64,92)(65,91)(66,90) );

G=PermutationGroup([[(1,92,65),(2,93,66),(3,94,34),(4,95,35),(5,96,36),(6,97,37),(7,98,38),(8,99,39),(9,67,40),(10,68,41),(11,69,42),(12,70,43),(13,71,44),(14,72,45),(15,73,46),(16,74,47),(17,75,48),(18,76,49),(19,77,50),(20,78,51),(21,79,52),(22,80,53),(23,81,54),(24,82,55),(25,83,56),(26,84,57),(27,85,58),(28,86,59),(29,87,60),(30,88,61),(31,89,62),(32,90,63),(33,91,64)], [(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)], [(1,33),(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,25),(10,24),(11,23),(12,22),(13,21),(14,20),(15,19),(16,18),(34,89),(35,88),(36,87),(37,86),(38,85),(39,84),(40,83),(41,82),(42,81),(43,80),(44,79),(45,78),(46,77),(47,76),(48,75),(49,74),(50,73),(51,72),(52,71),(53,70),(54,69),(55,68),(56,67),(57,99),(58,98),(59,97),(60,96),(61,95),(62,94),(63,93),(64,92),(65,91),(66,90)]])

C3⋊D33 is a maximal subgroup of   C3⋊S3×D11  S3×D33
C3⋊D33 is a maximal quotient of   C3⋊Dic33

51 conjugacy classes

 class 1 2 3A 3B 3C 3D 11A ··· 11E 33A ··· 33AN order 1 2 3 3 3 3 11 ··· 11 33 ··· 33 size 1 99 2 2 2 2 2 ··· 2 2 ··· 2

51 irreducible representations

 dim 1 1 2 2 2 type + + + + + image C1 C2 S3 D11 D33 kernel C3⋊D33 C3×C33 C33 C32 C3 # reps 1 1 4 5 40

Matrix representation of C3⋊D33 in GL4(𝔽67) generated by

 58 41 0 0 26 8 0 0 0 0 58 41 0 0 26 8
,
 31 57 0 0 10 53 0 0 0 0 64 52 0 0 15 30
,
 31 57 0 0 29 36 0 0 0 0 36 10 0 0 38 31
G:=sub<GL(4,GF(67))| [58,26,0,0,41,8,0,0,0,0,58,26,0,0,41,8],[31,10,0,0,57,53,0,0,0,0,64,15,0,0,52,30],[31,29,0,0,57,36,0,0,0,0,36,38,0,0,10,31] >;

C3⋊D33 in GAP, Magma, Sage, TeX

C_3\rtimes D_{33}
% in TeX

G:=Group("C3:D33");
// GroupNames label

G:=SmallGroup(198,9);
// by ID

G=gap.SmallGroup(198,9);
# by ID

G:=PCGroup([4,-2,-3,-3,-11,33,146,2883]);
// Polycyclic

G:=Group<a,b,c|a^3=b^33=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

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