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G = C4×C11⋊C5order 220 = 22·5·11

Direct product of C4 and C11⋊C5

direct product, metacyclic, supersoluble, monomial, Z-group, 5-hyperelementary

Aliases: C4×C11⋊C5, C44⋊C5, C112C20, C22.2C10, C2.(C2×C11⋊C5), (C2×C11⋊C5).2C2, SmallGroup(220,2)

Series: Derived Chief Lower central Upper central

C1C11 — C4×C11⋊C5
C1C11C22C2×C11⋊C5 — C4×C11⋊C5
C11 — C4×C11⋊C5
C1C4

Generators and relations for C4×C11⋊C5
 G = < a,b,c | a4=b11=c5=1, ab=ba, ac=ca, cbc-1=b3 >

11C5
11C10
11C20

Character table of C4×C11⋊C5

 class 124A4B5A5B5C5D10A10B10C10D11A11B20A20B20C20D20E20F20G20H22A22B44A44B44C44D
 size 11111111111111111111551111111111111111555555
ρ11111111111111111111111111111    trivial
ρ211-1-11111111111-1-1-1-1-1-1-1-111-1-1-1-1    linear of order 2
ρ31-1-ii1111-1-1-1-111i-ii-ii-ii-i-1-1-i-iii    linear of order 4
ρ41-1i-i1111-1-1-1-111-ii-ii-ii-ii-1-1ii-i-i    linear of order 4
ρ51111ζ52ζ53ζ5ζ54ζ5ζ53ζ54ζ5211ζ52ζ53ζ53ζ5ζ5ζ54ζ54ζ52111111    linear of order 5
ρ611-1-1ζ52ζ53ζ5ζ54ζ5ζ53ζ54ζ52115253535554545211-1-1-1-1    linear of order 10
ρ711-1-1ζ54ζ5ζ52ζ53ζ52ζ5ζ53ζ54115455525253535411-1-1-1-1    linear of order 10
ρ81111ζ5ζ54ζ53ζ52ζ53ζ54ζ52ζ511ζ5ζ54ζ54ζ53ζ53ζ52ζ52ζ5111111    linear of order 5
ρ911-1-1ζ5ζ54ζ53ζ52ζ53ζ54ζ52ζ5115545453535252511-1-1-1-1    linear of order 10
ρ101111ζ53ζ52ζ54ζ5ζ54ζ52ζ5ζ5311ζ53ζ52ζ52ζ54ζ54ζ5ζ5ζ53111111    linear of order 5
ρ1111-1-1ζ53ζ52ζ54ζ5ζ54ζ52ζ5ζ53115352525454555311-1-1-1-1    linear of order 10
ρ121111ζ54ζ5ζ52ζ53ζ52ζ5ζ53ζ5411ζ54ζ5ζ5ζ52ζ52ζ53ζ53ζ54111111    linear of order 5
ρ131-1-iiζ54ζ5ζ52ζ53525535411ζ4ζ54ζ43ζ5ζ4ζ5ζ43ζ52ζ4ζ52ζ43ζ53ζ4ζ53ζ43ζ54-1-1-i-iii    linear of order 20
ρ141-1-iiζ5ζ54ζ53ζ52535452511ζ4ζ5ζ43ζ54ζ4ζ54ζ43ζ53ζ4ζ53ζ43ζ52ζ4ζ52ζ43ζ5-1-1-i-iii    linear of order 20
ρ151-1i-iζ52ζ53ζ5ζ54553545211ζ43ζ52ζ4ζ53ζ43ζ53ζ4ζ5ζ43ζ5ζ4ζ54ζ43ζ54ζ4ζ52-1-1ii-i-i    linear of order 20
ρ161-1i-iζ53ζ52ζ54ζ5545255311ζ43ζ53ζ4ζ52ζ43ζ52ζ4ζ54ζ43ζ54ζ4ζ5ζ43ζ5ζ4ζ53-1-1ii-i-i    linear of order 20
ρ171-1i-iζ5ζ54ζ53ζ52535452511ζ43ζ5ζ4ζ54ζ43ζ54ζ4ζ53ζ43ζ53ζ4ζ52ζ43ζ52ζ4ζ5-1-1ii-i-i    linear of order 20
ρ181-1-iiζ52ζ53ζ5ζ54553545211ζ4ζ52ζ43ζ53ζ4ζ53ζ43ζ5ζ4ζ5ζ43ζ54ζ4ζ54ζ43ζ52-1-1-i-iii    linear of order 20
ρ191-1i-iζ54ζ5ζ52ζ53525535411ζ43ζ54ζ4ζ5ζ43ζ5ζ4ζ52ζ43ζ52ζ4ζ53ζ43ζ53ζ4ζ54-1-1ii-i-i    linear of order 20
ρ201-1-iiζ53ζ52ζ54ζ5545255311ζ4ζ53ζ43ζ52ζ4ζ52ζ43ζ54ζ4ζ54ζ43ζ5ζ4ζ5ζ43ζ53-1-1-i-iii    linear of order 20
ρ2155-5-500000000-1--11/2-1+-11/200000000-1--11/2-1+-11/21--11/21+-11/21+-11/21--11/2    complex lifted from C2×C11⋊C5
ρ2255-5-500000000-1+-11/2-1--11/200000000-1+-11/2-1--11/21+-11/21--11/21--11/21+-11/2    complex lifted from C2×C11⋊C5
ρ23555500000000-1+-11/2-1--11/200000000-1+-11/2-1--11/2-1--11/2-1+-11/2-1+-11/2-1--11/2    complex lifted from C11⋊C5
ρ24555500000000-1--11/2-1+-11/200000000-1--11/2-1+-11/2-1+-11/2-1--11/2-1--11/2-1+-11/2    complex lifted from C11⋊C5
ρ255-55i-5i00000000-1+-11/2-1--11/2000000001--11/21+-11/2ζ4ζ11104ζ1184ζ1174ζ1164ζ112ζ4ζ1194ζ1154ζ1144ζ1134ζ11ζ43ζ11943ζ11543ζ11443ζ11343ζ11ζ43ζ111043ζ11843ζ11743ζ11643ζ112    complex faithful
ρ265-55i-5i00000000-1--11/2-1+-11/2000000001+-11/21--11/2ζ4ζ1194ζ1154ζ1144ζ1134ζ11ζ4ζ11104ζ1184ζ1174ζ1164ζ112ζ43ζ111043ζ11843ζ11743ζ11643ζ112ζ43ζ11943ζ11543ζ11443ζ11343ζ11    complex faithful
ρ275-5-5i5i00000000-1+-11/2-1--11/2000000001--11/21+-11/2ζ43ζ111043ζ11843ζ11743ζ11643ζ112ζ43ζ11943ζ11543ζ11443ζ11343ζ11ζ4ζ1194ζ1154ζ1144ζ1134ζ11ζ4ζ11104ζ1184ζ1174ζ1164ζ112    complex faithful
ρ285-5-5i5i00000000-1--11/2-1+-11/2000000001+-11/21--11/2ζ43ζ11943ζ11543ζ11443ζ11343ζ11ζ43ζ111043ζ11843ζ11743ζ11643ζ112ζ4ζ11104ζ1184ζ1174ζ1164ζ112ζ4ζ1194ζ1154ζ1144ζ1134ζ11    complex faithful

Smallest permutation representation of C4×C11⋊C5
On 44 points
Generators in S44
(1 34 12 23)(2 35 13 24)(3 36 14 25)(4 37 15 26)(5 38 16 27)(6 39 17 28)(7 40 18 29)(8 41 19 30)(9 42 20 31)(10 43 21 32)(11 44 22 33)
(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)
(2 5 6 10 4)(3 9 11 8 7)(13 16 17 21 15)(14 20 22 19 18)(24 27 28 32 26)(25 31 33 30 29)(35 38 39 43 37)(36 42 44 41 40)

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

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

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

C4×C11⋊C5 is a maximal subgroup of   C11⋊C40  C4.F11  D44⋊C5

Matrix representation of C4×C11⋊C5 in GL6(𝔽661)

55500000
010000
001000
000100
000010
000001
,
100000
0432615441
0442615441
0433615441
0432616441
0432615451
,
24700000
000100
04561865946617
046617432616
010000
000010

G:=sub<GL(6,GF(661))| [555,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,43,44,43,43,43,0,2,2,3,2,2,0,615,615,615,616,615,0,44,44,44,44,45,0,1,1,1,1,1],[247,0,0,0,0,0,0,0,45,46,1,0,0,0,618,617,0,0,0,1,659,43,0,0,0,0,46,2,0,1,0,0,617,616,0,0] >;

C4×C11⋊C5 in GAP, Magma, Sage, TeX

C_4\times C_{11}\rtimes C_5
% in TeX

G:=Group("C4xC11:C5");
// GroupNames label

G:=SmallGroup(220,2);
// by ID

G=gap.SmallGroup(220,2);
# by ID

G:=PCGroup([4,-2,-5,-2,-11,40,647]);
// Polycyclic

G:=Group<a,b,c|a^4=b^11=c^5=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

Export

Subgroup lattice of C4×C11⋊C5 in TeX
Character table of C4×C11⋊C5 in TeX

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