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G = C11⋊S4  order 264 = 23·3·11

The semidirect product of C11 and S4 acting via S4/A4=C2

non-abelian, soluble, monomial

Aliases: C11⋊S4, A4⋊D11, C22⋊D33, (C2×C22)⋊2S3, (C11×A4)⋊1C2, SmallGroup(264,32)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C11×A4 — C11⋊S4
Chief series
C1 — C22 — C2×C22 — C11×A4 — C11⋊S4
Lower central
C11×A4 — C11⋊S4
Upper central
C1

Generators and relations for C11⋊S4
 G = < a,b,c,d,e | a11=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >

C1
3C2
66C2
4C3
C11
C22
33C22
33C4
44S3
3C22
6D11
4C33
33D4
A4
C2×C22
3D22
3Dic11
4D33
11S4
3C11⋊D4
C11×A4
C11⋊S4

Character table of C11⋊S4

 class 12A2B3411A11B11C11D11E22A22B22C22D22E33A33B33C33D33E33F33G33H33I33J
 size 136686622222666668888888888
ρ11111111111111111111111111    trivial
ρ211-11-111111111111111111111    linear of order 2
ρ3220-102222222222-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ422020ζ117+ζ114ζ118+ζ113ζ1110+ζ11ζ119+ζ112ζ116+ζ115ζ116+ζ115ζ1110+ζ11ζ117+ζ114ζ119+ζ112ζ118+ζ113ζ116+ζ115ζ118+ζ113ζ1110+ζ11ζ1110+ζ11ζ118+ζ113ζ116+ζ115ζ117+ζ114ζ119+ζ112ζ119+ζ112ζ117+ζ114    orthogonal lifted from D11
ρ522020ζ116+ζ115ζ1110+ζ11ζ117+ζ114ζ118+ζ113ζ119+ζ112ζ119+ζ112ζ117+ζ114ζ116+ζ115ζ118+ζ113ζ1110+ζ11ζ119+ζ112ζ1110+ζ11ζ117+ζ114ζ117+ζ114ζ1110+ζ11ζ119+ζ112ζ116+ζ115ζ118+ζ113ζ118+ζ113ζ116+ζ115    orthogonal lifted from D11
ρ622020ζ119+ζ112ζ117+ζ114ζ116+ζ115ζ1110+ζ11ζ118+ζ113ζ118+ζ113ζ116+ζ115ζ119+ζ112ζ1110+ζ11ζ117+ζ114ζ118+ζ113ζ117+ζ114ζ116+ζ115ζ116+ζ115ζ117+ζ114ζ118+ζ113ζ119+ζ112ζ1110+ζ11ζ1110+ζ11ζ119+ζ112    orthogonal lifted from D11
ρ722020ζ1110+ζ11ζ119+ζ112ζ118+ζ113ζ116+ζ115ζ117+ζ114ζ117+ζ114ζ118+ζ113ζ1110+ζ11ζ116+ζ115ζ119+ζ112ζ117+ζ114ζ119+ζ112ζ118+ζ113ζ118+ζ113ζ119+ζ112ζ117+ζ114ζ1110+ζ11ζ116+ζ115ζ116+ζ115ζ1110+ζ11    orthogonal lifted from D11
ρ822020ζ118+ζ113ζ116+ζ115ζ119+ζ112ζ117+ζ114ζ1110+ζ11ζ1110+ζ11ζ119+ζ112ζ118+ζ113ζ117+ζ114ζ116+ζ115ζ1110+ζ11ζ116+ζ115ζ119+ζ112ζ119+ζ112ζ116+ζ115ζ1110+ζ11ζ118+ζ113ζ117+ζ114ζ117+ζ114ζ118+ζ113    orthogonal lifted from D11
ρ9220-10ζ1110+ζ11ζ119+ζ112ζ118+ζ113ζ116+ζ115ζ117+ζ114ζ117+ζ114ζ118+ζ113ζ1110+ζ11ζ116+ζ115ζ119+ζ112-ζ3ζ117+ζ3ζ114-ζ117ζ3ζ119-ζ3ζ112-ζ112ζ32ζ118-ζ32ζ113-ζ113ζ3ζ118-ζ3ζ113-ζ113ζ32ζ119-ζ32ζ112-ζ112ζ3ζ117-ζ3ζ114-ζ114ζ32ζ1110-ζ32ζ11-ζ11ζ3ζ116-ζ3ζ115-ζ115-ζ3ζ116+ζ3ζ115-ζ116-ζ32ζ1110+ζ32ζ11-ζ1110    orthogonal lifted from D33
ρ10220-10ζ116+ζ115ζ1110+ζ11ζ117+ζ114ζ118+ζ113ζ119+ζ112ζ119+ζ112ζ117+ζ114ζ116+ζ115ζ118+ζ113ζ1110+ζ11ζ3ζ119-ζ3ζ112-ζ112ζ32ζ1110-ζ32ζ11-ζ11-ζ3ζ117+ζ3ζ114-ζ117ζ3ζ117-ζ3ζ114-ζ114-ζ32ζ1110+ζ32ζ11-ζ1110ζ32ζ119-ζ32ζ112-ζ112-ζ3ζ116+ζ3ζ115-ζ116ζ3ζ118-ζ3ζ113-ζ113ζ32ζ118-ζ32ζ113-ζ113ζ3ζ116-ζ3ζ115-ζ115    orthogonal lifted from D33
ρ11220-10ζ1110+ζ11ζ119+ζ112ζ118+ζ113ζ116+ζ115ζ117+ζ114ζ117+ζ114ζ118+ζ113ζ1110+ζ11ζ116+ζ115ζ119+ζ112ζ3ζ117-ζ3ζ114-ζ114ζ32ζ119-ζ32ζ112-ζ112ζ3ζ118-ζ3ζ113-ζ113ζ32ζ118-ζ32ζ113-ζ113ζ3ζ119-ζ3ζ112-ζ112-ζ3ζ117+ζ3ζ114-ζ117-ζ32ζ1110+ζ32ζ11-ζ1110-ζ3ζ116+ζ3ζ115-ζ116ζ3ζ116-ζ3ζ115-ζ115ζ32ζ1110-ζ32ζ11-ζ11    orthogonal lifted from D33
ρ12220-10ζ118+ζ113ζ116+ζ115ζ119+ζ112ζ117+ζ114ζ1110+ζ11ζ1110+ζ11ζ119+ζ112ζ118+ζ113ζ117+ζ114ζ116+ζ115-ζ32ζ1110+ζ32ζ11-ζ1110ζ3ζ116-ζ3ζ115-ζ115ζ32ζ119-ζ32ζ112-ζ112ζ3ζ119-ζ3ζ112-ζ112-ζ3ζ116+ζ3ζ115-ζ116ζ32ζ1110-ζ32ζ11-ζ11ζ3ζ118-ζ3ζ113-ζ113-ζ3ζ117+ζ3ζ114-ζ117ζ3ζ117-ζ3ζ114-ζ114ζ32ζ118-ζ32ζ113-ζ113    orthogonal lifted from D33
ρ13220-10ζ117+ζ114ζ118+ζ113ζ1110+ζ11ζ119+ζ112ζ116+ζ115ζ116+ζ115ζ1110+ζ11ζ117+ζ114ζ119+ζ112ζ118+ζ113-ζ3ζ116+ζ3ζ115-ζ116ζ32ζ118-ζ32ζ113-ζ113ζ32ζ1110-ζ32ζ11-ζ11-ζ32ζ1110+ζ32ζ11-ζ1110ζ3ζ118-ζ3ζ113-ζ113ζ3ζ116-ζ3ζ115-ζ115-ζ3ζ117+ζ3ζ114-ζ117ζ32ζ119-ζ32ζ112-ζ112ζ3ζ119-ζ3ζ112-ζ112ζ3ζ117-ζ3ζ114-ζ114    orthogonal lifted from D33
ρ14220-10ζ119+ζ112ζ117+ζ114ζ116+ζ115ζ1110+ζ11ζ118+ζ113ζ118+ζ113ζ116+ζ115ζ119+ζ112ζ1110+ζ11ζ117+ζ114ζ32ζ118-ζ32ζ113-ζ113-ζ3ζ117+ζ3ζ114-ζ117-ζ3ζ116+ζ3ζ115-ζ116ζ3ζ116-ζ3ζ115-ζ115ζ3ζ117-ζ3ζ114-ζ114ζ3ζ118-ζ3ζ113-ζ113ζ3ζ119-ζ3ζ112-ζ112-ζ32ζ1110+ζ32ζ11-ζ1110ζ32ζ1110-ζ32ζ11-ζ11ζ32ζ119-ζ32ζ112-ζ112    orthogonal lifted from D33
ρ15220-10ζ118+ζ113ζ116+ζ115ζ119+ζ112ζ117+ζ114ζ1110+ζ11ζ1110+ζ11ζ119+ζ112ζ118+ζ113ζ117+ζ114ζ116+ζ115ζ32ζ1110-ζ32ζ11-ζ11-ζ3ζ116+ζ3ζ115-ζ116ζ3ζ119-ζ3ζ112-ζ112ζ32ζ119-ζ32ζ112-ζ112ζ3ζ116-ζ3ζ115-ζ115-ζ32ζ1110+ζ32ζ11-ζ1110ζ32ζ118-ζ32ζ113-ζ113ζ3ζ117-ζ3ζ114-ζ114-ζ3ζ117+ζ3ζ114-ζ117ζ3ζ118-ζ3ζ113-ζ113    orthogonal lifted from D33
ρ16220-10ζ117+ζ114ζ118+ζ113ζ1110+ζ11ζ119+ζ112ζ116+ζ115ζ116+ζ115ζ1110+ζ11ζ117+ζ114ζ119+ζ112ζ118+ζ113ζ3ζ116-ζ3ζ115-ζ115ζ3ζ118-ζ3ζ113-ζ113-ζ32ζ1110+ζ32ζ11-ζ1110ζ32ζ1110-ζ32ζ11-ζ11ζ32ζ118-ζ32ζ113-ζ113-ζ3ζ116+ζ3ζ115-ζ116ζ3ζ117-ζ3ζ114-ζ114ζ3ζ119-ζ3ζ112-ζ112ζ32ζ119-ζ32ζ112-ζ112-ζ3ζ117+ζ3ζ114-ζ117    orthogonal lifted from D33
ρ17220-10ζ119+ζ112ζ117+ζ114ζ116+ζ115ζ1110+ζ11ζ118+ζ113ζ118+ζ113ζ116+ζ115ζ119+ζ112ζ1110+ζ11ζ117+ζ114ζ3ζ118-ζ3ζ113-ζ113ζ3ζ117-ζ3ζ114-ζ114ζ3ζ116-ζ3ζ115-ζ115-ζ3ζ116+ζ3ζ115-ζ116-ζ3ζ117+ζ3ζ114-ζ117ζ32ζ118-ζ32ζ113-ζ113ζ32ζ119-ζ32ζ112-ζ112ζ32ζ1110-ζ32ζ11-ζ11-ζ32ζ1110+ζ32ζ11-ζ1110ζ3ζ119-ζ3ζ112-ζ112    orthogonal lifted from D33
ρ18220-10ζ116+ζ115ζ1110+ζ11ζ117+ζ114ζ118+ζ113ζ119+ζ112ζ119+ζ112ζ117+ζ114ζ116+ζ115ζ118+ζ113ζ1110+ζ11ζ32ζ119-ζ32ζ112-ζ112-ζ32ζ1110+ζ32ζ11-ζ1110ζ3ζ117-ζ3ζ114-ζ114-ζ3ζ117+ζ3ζ114-ζ117ζ32ζ1110-ζ32ζ11-ζ11ζ3ζ119-ζ3ζ112-ζ112ζ3ζ116-ζ3ζ115-ζ115ζ32ζ118-ζ32ζ113-ζ113ζ3ζ118-ζ3ζ113-ζ113-ζ3ζ116+ζ3ζ115-ζ116    orthogonal lifted from D33
ρ193-1-10133333-1-1-1-1-10000000000    orthogonal lifted from S4
ρ203-110-133333-1-1-1-1-10000000000    orthogonal lifted from S4
ρ216-20003ζ118+3ζ1133ζ116+3ζ1153ζ119+3ζ1123ζ117+3ζ1143ζ1110+3ζ11-ζ1110-ζ11-ζ119-ζ112-ζ118-ζ113-ζ117-ζ114-ζ116-ζ1150000000000    orthogonal faithful
ρ226-20003ζ1110+3ζ113ζ119+3ζ1123ζ118+3ζ1133ζ116+3ζ1153ζ117+3ζ114-ζ117-ζ114-ζ118-ζ113-ζ1110-ζ11-ζ116-ζ115-ζ119-ζ1120000000000    orthogonal faithful
ρ236-20003ζ117+3ζ1143ζ118+3ζ1133ζ1110+3ζ113ζ119+3ζ1123ζ116+3ζ115-ζ116-ζ115-ζ1110-ζ11-ζ117-ζ114-ζ119-ζ112-ζ118-ζ1130000000000    orthogonal faithful
ρ246-20003ζ116+3ζ1153ζ1110+3ζ113ζ117+3ζ1143ζ118+3ζ1133ζ119+3ζ112-ζ119-ζ112-ζ117-ζ114-ζ116-ζ115-ζ118-ζ113-ζ1110-ζ110000000000    orthogonal faithful
ρ256-20003ζ119+3ζ1123ζ117+3ζ1143ζ116+3ζ1153ζ1110+3ζ113ζ118+3ζ113-ζ118-ζ113-ζ116-ζ115-ζ119-ζ112-ζ1110-ζ11-ζ117-ζ1140000000000    orthogonal faithful

Smallest permutation representation of C11⋊S4
►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 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 12)(11 13)(23 41)(24 42)(25 43)(26 44)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)(33 40)

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

(12 38 31)(13 39 32)(14 40 33)(15 41 23)(16 42 24)(17 43 25)(18 44 26)(19 34 27)(20 35 28)(21 36 29)(22 37 30)

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

Matrix representation of C11⋊S4 ►in GL5(𝔽397)

267385000
12279000
00100
00010
00001
,
10000
01000
00011
0003960
00110
,
10000
01000
000396396
0039601
0000396
,
396396000
10000
003963960
00100
0039601
,
12130000
118385000
00010
00100
0000396

G:=sub<GL(5,GF(397))| [267,12,0,0,0,385,279,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,396,1,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,396,0,0,0,396,0,0,0,0,396,1,396],[396,1,0,0,0,396,0,0,0,0,0,0,396,1,396,0,0,396,0,0,0,0,0,0,1],[12,118,0,0,0,130,385,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,396] >;
 

C11⋊S4 in GAP, Magma, Sage, TeX 

C_{11}\rtimes S_4
 
% in TeX
 
G:=Group("C11:S4");
 
// GroupNames label
 
G:=SmallGroup(264,32);
 
// by ID
 
G=gap.SmallGroup(264,32);
 
# by ID
 
G:=PCGroup([5,-2,-3,-11,-2,2,41,902,2643,1328,1654,2484]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^11=b^2=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,d*b*d^-1=e*b*e=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C11⋊S4 in TeX
Character table of C11⋊S4 in TeX

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