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## G = C11×He3order 297 = 33·11

### Direct product of C11 and He3

direct product, metabelian, nilpotent (class 2), monomial, 3-elementary

Aliases: C11×He3, C32⋊C33, C33.1C32, (C3×C33)⋊C3, C3.1(C3×C33), SmallGroup(297,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C11×He3
 Chief series C1 — C3 — C33 — C3×C33 — C11×He3
 Lower central C1 — C3 — C11×He3
 Upper central C1 — C33 — C11×He3

Generators and relations for C11×He3
G = < a,b,c,d | a11=b3=c3=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, cd=dc >

Smallest permutation representation of C11×He3
On 99 points
Generators in S99
(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)
(12 94 33)(13 95 23)(14 96 24)(15 97 25)(16 98 26)(17 99 27)(18 89 28)(19 90 29)(20 91 30)(21 92 31)(22 93 32)(56 69 86)(57 70 87)(58 71 88)(59 72 78)(60 73 79)(61 74 80)(62 75 81)(63 76 82)(64 77 83)(65 67 84)(66 68 85)
(1 55 38)(2 45 39)(3 46 40)(4 47 41)(5 48 42)(6 49 43)(7 50 44)(8 51 34)(9 52 35)(10 53 36)(11 54 37)(12 33 94)(13 23 95)(14 24 96)(15 25 97)(16 26 98)(17 27 99)(18 28 89)(19 29 90)(20 30 91)(21 31 92)(22 32 93)(56 69 86)(57 70 87)(58 71 88)(59 72 78)(60 73 79)(61 74 80)(62 75 81)(63 76 82)(64 77 83)(65 67 84)(66 68 85)
(1 20 71)(2 21 72)(3 22 73)(4 12 74)(5 13 75)(6 14 76)(7 15 77)(8 16 67)(9 17 68)(10 18 69)(11 19 70)(23 81 48)(24 82 49)(25 83 50)(26 84 51)(27 85 52)(28 86 53)(29 87 54)(30 88 55)(31 78 45)(32 79 46)(33 80 47)(34 98 65)(35 99 66)(36 89 56)(37 90 57)(38 91 58)(39 92 59)(40 93 60)(41 94 61)(42 95 62)(43 96 63)(44 97 64)

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

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

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

121 conjugacy classes

 class 1 3A 3B 3C ··· 3J 11A ··· 11J 33A ··· 33T 33U ··· 33CV order 1 3 3 3 ··· 3 11 ··· 11 33 ··· 33 33 ··· 33 size 1 1 1 3 ··· 3 1 ··· 1 1 ··· 1 3 ··· 3

121 irreducible representations

 dim 1 1 1 1 3 3 type + image C1 C3 C11 C33 He3 C11×He3 kernel C11×He3 C3×C33 He3 C32 C11 C1 # reps 1 8 10 80 2 20

Matrix representation of C11×He3 in GL3(𝔽67) generated by

 22 0 0 0 22 0 0 0 22
,
 1 0 0 46 37 0 23 0 29
,
 37 0 0 0 37 0 0 0 37
,
 46 36 0 46 21 1 21 47 0
G:=sub<GL(3,GF(67))| [22,0,0,0,22,0,0,0,22],[1,46,23,0,37,0,0,0,29],[37,0,0,0,37,0,0,0,37],[46,46,21,36,21,47,0,1,0] >;

C11×He3 in GAP, Magma, Sage, TeX

C_{11}\times {\rm He}_3
% in TeX

G:=Group("C11xHe3");
// GroupNames label

G:=SmallGroup(297,3);
// by ID

G=gap.SmallGroup(297,3);
# by ID

G:=PCGroup([4,-3,-3,-11,-3,817]);
// Polycyclic

G:=Group<a,b,c,d|a^11=b^3=c^3=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^-1,c*d=d*c>;
// generators/relations

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