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## G = C2×S3×D11order 264 = 23·3·11

### Direct product of C2, S3 and D11

Aliases: C2×S3×D11, C33⋊C23, C221D6, C61D22, C66⋊C22, D665C2, D33⋊C22, (S3×C22)⋊3C2, (C6×D11)⋊3C2, (S3×C11)⋊C22, C111(C22×S3), (C3×D11)⋊C22, C31(C22×D11), SmallGroup(264,34)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C2×S3×D11
 Chief series C1 — C11 — C33 — C3×D11 — S3×D11 — C2×S3×D11
 Lower central C33 — C2×S3×D11
 Upper central C1 — C2

Generators and relations for C2×S3×D11
G = < a,b,c,d,e | a2=b3=c2=d11=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 488 in 64 conjugacy classes, 28 normal (14 characteristic)
C1, C2, C2, C3, C22, S3, S3, C6, C6, C23, C11, D6, D6, C2×C6, D11, D11, C22, C22, C22×S3, C33, D22, D22, C2×C22, S3×C11, C3×D11, D33, C66, C22×D11, S3×D11, C6×D11, S3×C22, D66, C2×S3×D11
Quotients: C1, C2, C22, S3, C23, D6, D11, C22×S3, D22, C22×D11, S3×D11, C2×S3×D11

Smallest permutation representation of C2×S3×D11
On 66 points
Generators in S66
(1 43)(2 44)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 45)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 57)(25 58)(26 59)(27 60)(28 61)(29 62)(30 63)(31 64)(32 65)(33 66)
(1 21 32)(2 22 33)(3 12 23)(4 13 24)(5 14 25)(6 15 26)(7 16 27)(8 17 28)(9 18 29)(10 19 30)(11 20 31)(34 45 56)(35 46 57)(36 47 58)(37 48 59)(38 49 60)(39 50 61)(40 51 62)(41 52 63)(42 53 64)(43 54 65)(44 55 66)
(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 31)(21 32)(22 33)(45 56)(46 57)(47 58)(48 59)(49 60)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)
(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)
(1 11)(2 10)(3 9)(4 8)(5 7)(12 18)(13 17)(14 16)(19 22)(20 21)(23 29)(24 28)(25 27)(30 33)(31 32)(34 40)(35 39)(36 38)(41 44)(42 43)(45 51)(46 50)(47 49)(52 55)(53 54)(56 62)(57 61)(58 60)(63 66)(64 65)

G:=sub<Sym(66)| (1,43)(2,44)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,65)(33,66), (1,21,32)(2,22,33)(3,12,23)(4,13,24)(5,14,25)(6,15,26)(7,16,27)(8,17,28)(9,18,29)(10,19,30)(11,20,31)(34,45,56)(35,46,57)(36,47,58)(37,48,59)(38,49,60)(39,50,61)(40,51,62)(41,52,63)(42,53,64)(43,54,65)(44,55,66), (12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33)(45,56)(46,57)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66), (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), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)(34,40)(35,39)(36,38)(41,44)(42,43)(45,51)(46,50)(47,49)(52,55)(53,54)(56,62)(57,61)(58,60)(63,66)(64,65)>;

G:=Group( (1,43)(2,44)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,65)(33,66), (1,21,32)(2,22,33)(3,12,23)(4,13,24)(5,14,25)(6,15,26)(7,16,27)(8,17,28)(9,18,29)(10,19,30)(11,20,31)(34,45,56)(35,46,57)(36,47,58)(37,48,59)(38,49,60)(39,50,61)(40,51,62)(41,52,63)(42,53,64)(43,54,65)(44,55,66), (12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33)(45,56)(46,57)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66), (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), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)(34,40)(35,39)(36,38)(41,44)(42,43)(45,51)(46,50)(47,49)(52,55)(53,54)(56,62)(57,61)(58,60)(63,66)(64,65) );

G=PermutationGroup([[(1,43),(2,44),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,41),(11,42),(12,45),(13,46),(14,47),(15,48),(16,49),(17,50),(18,51),(19,52),(20,53),(21,54),(22,55),(23,56),(24,57),(25,58),(26,59),(27,60),(28,61),(29,62),(30,63),(31,64),(32,65),(33,66)], [(1,21,32),(2,22,33),(3,12,23),(4,13,24),(5,14,25),(6,15,26),(7,16,27),(8,17,28),(9,18,29),(10,19,30),(11,20,31),(34,45,56),(35,46,57),(36,47,58),(37,48,59),(38,49,60),(39,50,61),(40,51,62),(41,52,63),(42,53,64),(43,54,65),(44,55,66)], [(12,23),(13,24),(14,25),(15,26),(16,27),(17,28),(18,29),(19,30),(20,31),(21,32),(22,33),(45,56),(46,57),(47,58),(48,59),(49,60),(50,61),(51,62),(52,63),(53,64),(54,65),(55,66)], [(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)], [(1,11),(2,10),(3,9),(4,8),(5,7),(12,18),(13,17),(14,16),(19,22),(20,21),(23,29),(24,28),(25,27),(30,33),(31,32),(34,40),(35,39),(36,38),(41,44),(42,43),(45,51),(46,50),(47,49),(52,55),(53,54),(56,62),(57,61),(58,60),(63,66),(64,65)]])

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 6A 6B 6C 11A ··· 11E 22A ··· 22E 22F ··· 22O 33A ··· 33E 66A ··· 66E order 1 2 2 2 2 2 2 2 3 6 6 6 11 ··· 11 22 ··· 22 22 ··· 22 33 ··· 33 66 ··· 66 size 1 1 3 3 11 11 33 33 2 2 22 22 2 ··· 2 2 ··· 2 6 ··· 6 4 ··· 4 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D6 D6 D11 D22 D22 S3×D11 C2×S3×D11 kernel C2×S3×D11 S3×D11 C6×D11 S3×C22 D66 D22 D11 C22 D6 S3 C6 C2 C1 # reps 1 4 1 1 1 1 2 1 5 10 5 5 5

Matrix representation of C2×S3×D11 in GL4(𝔽67) generated by

 1 0 0 0 0 1 0 0 0 0 66 0 0 0 0 66
,
 1 0 0 0 0 1 0 0 0 0 65 17 0 0 55 1
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 12 66
,
 0 1 0 0 66 19 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(67))| [1,0,0,0,0,1,0,0,0,0,66,0,0,0,0,66],[1,0,0,0,0,1,0,0,0,0,65,55,0,0,17,1],[1,0,0,0,0,1,0,0,0,0,1,12,0,0,0,66],[0,66,0,0,1,19,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;

C2×S3×D11 in GAP, Magma, Sage, TeX

C_2\times S_3\times D_{11}
% in TeX

G:=Group("C2xS3xD11");
// GroupNames label

G:=SmallGroup(264,34);
// by ID

G=gap.SmallGroup(264,34);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-11,168,6004]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^11=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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