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G = D4×C11order 88 = 23·11

Direct product of C11 and D4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: D4×C11, C4⋊C22, C443C2, C22⋊C22, C22.6C22, (C2×C22)⋊1C2, C2.1(C2×C22), SmallGroup(88,9)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C11
C1C2C22C2×C22 — D4×C11
C1C2 — D4×C11
C1C22 — D4×C11

Generators and relations for D4×C11
 G = < a,b,c | a11=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

2C2
2C2
2C22
2C22

Smallest permutation representation of D4×C11
On 44 points
Generators in S44
(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)
(1 38 31 20)(2 39 32 21)(3 40 33 22)(4 41 23 12)(5 42 24 13)(6 43 25 14)(7 44 26 15)(8 34 27 16)(9 35 28 17)(10 36 29 18)(11 37 30 19)
(12 41)(13 42)(14 43)(15 44)(16 34)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)

G:=sub<Sym(44)| (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), (1,38,31,20)(2,39,32,21)(3,40,33,22)(4,41,23,12)(5,42,24,13)(6,43,25,14)(7,44,26,15)(8,34,27,16)(9,35,28,17)(10,36,29,18)(11,37,30,19), (12,41)(13,42)(14,43)(15,44)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)>;

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), (1,38,31,20)(2,39,32,21)(3,40,33,22)(4,41,23,12)(5,42,24,13)(6,43,25,14)(7,44,26,15)(8,34,27,16)(9,35,28,17)(10,36,29,18)(11,37,30,19), (12,41)(13,42)(14,43)(15,44)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40) );

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)], [(1,38,31,20),(2,39,32,21),(3,40,33,22),(4,41,23,12),(5,42,24,13),(6,43,25,14),(7,44,26,15),(8,34,27,16),(9,35,28,17),(10,36,29,18),(11,37,30,19)], [(12,41),(13,42),(14,43),(15,44),(16,34),(17,35),(18,36),(19,37),(20,38),(21,39),(22,40)])

D4×C11 is a maximal subgroup of   D4⋊D11  D4.D11  D42D11

55 conjugacy classes

class 1 2A2B2C 4 11A···11J22A···22J22K···22AD44A···44J
order1222411···1122···2222···2244···44
size112221···11···12···22···2

55 irreducible representations

dim11111122
type++++
imageC1C2C2C11C22C22D4D4×C11
kernelD4×C11C44C2×C22D4C4C22C11C1
# reps112101020110

Matrix representation of D4×C11 in GL2(𝔽23) generated by

180
018
,
08
200
,
015
200
G:=sub<GL(2,GF(23))| [18,0,0,18],[0,20,8,0],[0,20,15,0] >;

D4×C11 in GAP, Magma, Sage, TeX

D_4\times C_{11}
% in TeX

G:=Group("D4xC11");
// GroupNames label

G:=SmallGroup(88,9);
// by ID

G=gap.SmallGroup(88,9);
# by ID

G:=PCGroup([4,-2,-2,-11,-2,369]);
// Polycyclic

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

Export

Subgroup lattice of D4×C11 in TeX

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