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G = C11×C4≀C2order 352 = 25·11

Direct product of C11 and C4≀C2

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

Aliases: C11×C4≀C2, D42C44, Q82C44, C423C22, C44.66D4, M4(2)⋊4C22, (C4×C44)⋊10C2, (D4×C11)⋊5C4, C4.3(C2×C44), (Q8×C11)⋊5C4, C44.30(C2×C4), C4○D4.1C22, C4.17(D4×C11), (C2×C22).22D4, C22.3(D4×C11), C22.26(C22⋊C4), (C11×M4(2))⋊10C2, (C2×C44).116C22, (C2×C4).19(C2×C22), (C11×C4○D4).4C2, C2.8(C11×C22⋊C4), SmallGroup(352,53)

Series: Derived Chief Lower central Upper central

C1C4 — C11×C4≀C2
C1C2C4C2×C4C2×C44C11×M4(2) — C11×C4≀C2
C1C2C4 — C11×C4≀C2
C1C44C2×C44 — C11×C4≀C2

Generators and relations for C11×C4≀C2
 G = < a,b,c,d | a11=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b-1c >

2C2
4C2
2C4
2C22
2C4
2C4
2C22
4C22
2D4
2C2×C4
2C2×C4
2C8
2C2×C22
2C44
2C44
2C44
2C2×C44
2C88
2D4×C11
2C2×C44

Smallest permutation representation of C11×C4≀C2
On 88 points
Generators in S88
(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)
(1 47 61 75)(2 48 62 76)(3 49 63 77)(4 50 64 67)(5 51 65 68)(6 52 66 69)(7 53 56 70)(8 54 57 71)(9 55 58 72)(10 45 59 73)(11 46 60 74)(12 34 24 84)(13 35 25 85)(14 36 26 86)(15 37 27 87)(16 38 28 88)(17 39 29 78)(18 40 30 79)(19 41 31 80)(20 42 32 81)(21 43 33 82)(22 44 23 83)
(1 40)(2 41)(3 42)(4 43)(5 44)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 52)(13 53)(14 54)(15 55)(16 45)(17 46)(18 47)(19 48)(20 49)(21 50)(22 51)(23 68)(24 69)(25 70)(26 71)(27 72)(28 73)(29 74)(30 75)(31 76)(32 77)(33 67)(56 85)(57 86)(58 87)(59 88)(60 78)(61 79)(62 80)(63 81)(64 82)(65 83)(66 84)
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 56)(8 57)(9 58)(10 59)(11 60)(12 34 24 84)(13 35 25 85)(14 36 26 86)(15 37 27 87)(16 38 28 88)(17 39 29 78)(18 40 30 79)(19 41 31 80)(20 42 32 81)(21 43 33 82)(22 44 23 83)(45 73)(46 74)(47 75)(48 76)(49 77)(50 67)(51 68)(52 69)(53 70)(54 71)(55 72)

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

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

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

154 conjugacy classes

class 1 2A2B2C4A4B4C···4G4H8A8B11A···11J22A···22J22K···22T22U···22AD44A···44T44U···44BR44BS···44CB88A···88T
order1222444···448811···1122···2222···2222···2244···4444···4444···4488···88
size1124112···24441···11···12···24···41···12···24···44···4

154 irreducible representations

dim111111111111222222
type++++++
imageC1C2C2C2C4C4C11C22C22C22C44C44D4D4C4≀C2D4×C11D4×C11C11×C4≀C2
kernelC11×C4≀C2C4×C44C11×M4(2)C11×C4○D4D4×C11Q8×C11C4≀C2C42M4(2)C4○D4D4Q8C44C2×C22C11C4C22C1
# reps111122101010102020114101040

Matrix representation of C11×C4≀C2 in GL2(𝔽89) generated by

450
045
,
340
055
,
055
340
,
880
055
G:=sub<GL(2,GF(89))| [45,0,0,45],[34,0,0,55],[0,34,55,0],[88,0,0,55] >;

C11×C4≀C2 in GAP, Magma, Sage, TeX

C_{11}\times C_4\wr C_2
% in TeX

G:=Group("C11xC4wrC2");
// GroupNames label

G:=SmallGroup(352,53);
// by ID

G=gap.SmallGroup(352,53);
# by ID

G:=PCGroup([6,-2,-2,-11,-2,-2,-2,528,553,5283,2649,117,88]);
// Polycyclic

G:=Group<a,b,c,d|a^11=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^-1*c>;
// generators/relations

Export

Subgroup lattice of C11×C4≀C2 in TeX

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