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G = C112⋊C4order 484 = 22·112

The semidirect product of C112 and C4 acting faithfully

metabelian, soluble, monomial, A-group

Aliases: C112⋊C4, C11⋊D11.C2, SmallGroup(484,8)

Series: Derived Chief Lower central Upper central

C1C112 — C112⋊C4
C1C112C11⋊D11 — C112⋊C4
C112 — C112⋊C4
C1

Generators and relations for C112⋊C4
 G = < a,b,c | a11=b11=c4=1, ab=ba, cac-1=a3b6, cbc-1=a2b8 >

121C2
2C11
2C11
2C11
2C11
2C11
2C11
121C4
22D11
22D11
22D11
22D11
22D11
22D11

Permutation representations of C112⋊C4
On 22 points - transitive group 22T8
Generators in S22
(1 2 3 4 5 6 7 8 9 10 11)(12 13 14 15 16 17 18 19 20 21 22)
(1 5 9 2 6 10 3 7 11 4 8)(12 21 19 17 15 13 22 20 18 16 14)
(1 14)(2 12 11 16)(3 21 10 18)(4 19 9 20)(5 17 8 22)(6 15 7 13)

G:=sub<Sym(22)| (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22), (1,5,9,2,6,10,3,7,11,4,8)(12,21,19,17,15,13,22,20,18,16,14), (1,14)(2,12,11,16)(3,21,10,18)(4,19,9,20)(5,17,8,22)(6,15,7,13)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22), (1,5,9,2,6,10,3,7,11,4,8)(12,21,19,17,15,13,22,20,18,16,14), (1,14)(2,12,11,16)(3,21,10,18)(4,19,9,20)(5,17,8,22)(6,15,7,13) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11),(12,13,14,15,16,17,18,19,20,21,22)], [(1,5,9,2,6,10,3,7,11,4,8),(12,21,19,17,15,13,22,20,18,16,14)], [(1,14),(2,12,11,16),(3,21,10,18),(4,19,9,20),(5,17,8,22),(6,15,7,13)])

G:=TransitiveGroup(22,8);

34 conjugacy classes

class 1  2 4A4B11A···11AD
order124411···11
size11211211214···4

34 irreducible representations

dim1114
type+++
imageC1C2C4C112⋊C4
kernelC112⋊C4C11⋊D11C112C1
# reps11230

Matrix representation of C112⋊C4 in GL4(𝔽89) generated by

185600
337800
005311
007833
,
0100
887100
007833
005618
,
0010
0001
1000
718800
G:=sub<GL(4,GF(89))| [18,33,0,0,56,78,0,0,0,0,53,78,0,0,11,33],[0,88,0,0,1,71,0,0,0,0,78,56,0,0,33,18],[0,0,1,71,0,0,0,88,1,0,0,0,0,1,0,0] >;

C112⋊C4 in GAP, Magma, Sage, TeX

C_{11}^2\rtimes C_4
% in TeX

G:=Group("C11^2:C4");
// GroupNames label

G:=SmallGroup(484,8);
// by ID

G=gap.SmallGroup(484,8);
# by ID

G:=PCGroup([4,-2,-2,-11,11,8,3026,246,2691,3527]);
// Polycyclic

G:=Group<a,b,c|a^11=b^11=c^4=1,a*b=b*a,c*a*c^-1=a^3*b^6,c*b*c^-1=a^2*b^8>;
// generators/relations

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Subgroup lattice of C112⋊C4 in TeX

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