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## G = C112⋊C3order 363 = 3·112

### The semidirect product of C112 and C3 acting faithfully

Aliases: C112⋊C3, SmallGroup(363,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C112 — C112⋊C3
 Chief series C1 — C112 — C112⋊C3
 Lower central C112 — C112⋊C3
 Upper central C1

Generators and relations for C112⋊C3
G = < a,b,c | a11=b11=c3=1, ab=ba, cac-1=a3b, cbc-1=a9b7 >

121C3
3C11
3C11
3C11
3C11

Smallest permutation representation of C112⋊C3
On 33 points
Generators in S33
(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)
(1 2 3 4 5 6 7 8 9 10 11)(12 22 21 20 19 18 17 16 15 14 13)(23 27 31 24 28 32 25 29 33 26 30)
(1 30 20)(2 33 22)(3 25 13)(4 28 15)(5 31 17)(6 23 19)(7 26 21)(8 29 12)(9 32 14)(10 24 16)(11 27 18)

G:=sub<Sym(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), (1,2,3,4,5,6,7,8,9,10,11)(12,22,21,20,19,18,17,16,15,14,13)(23,27,31,24,28,32,25,29,33,26,30), (1,30,20)(2,33,22)(3,25,13)(4,28,15)(5,31,17)(6,23,19)(7,26,21)(8,29,12)(9,32,14)(10,24,16)(11,27,18)>;

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), (1,2,3,4,5,6,7,8,9,10,11)(12,22,21,20,19,18,17,16,15,14,13)(23,27,31,24,28,32,25,29,33,26,30), (1,30,20)(2,33,22)(3,25,13)(4,28,15)(5,31,17)(6,23,19)(7,26,21)(8,29,12)(9,32,14)(10,24,16)(11,27,18) );

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)], [(1,2,3,4,5,6,7,8,9,10,11),(12,22,21,20,19,18,17,16,15,14,13),(23,27,31,24,28,32,25,29,33,26,30)], [(1,30,20),(2,33,22),(3,25,13),(4,28,15),(5,31,17),(6,23,19),(7,26,21),(8,29,12),(9,32,14),(10,24,16),(11,27,18)]])

43 conjugacy classes

 class 1 3A 3B 11A ··· 11AN order 1 3 3 11 ··· 11 size 1 121 121 3 ··· 3

43 irreducible representations

 dim 1 1 3 type + image C1 C3 C112⋊C3 kernel C112⋊C3 C112 C1 # reps 1 2 40

Matrix representation of C112⋊C3 in GL3(𝔽67) generated by

 14 0 0 0 9 0 0 0 25
,
 14 0 0 0 15 0 0 0 15
,
 0 1 0 0 0 1 1 0 0
G:=sub<GL(3,GF(67))| [14,0,0,0,9,0,0,0,25],[14,0,0,0,15,0,0,0,15],[0,0,1,1,0,0,0,1,0] >;

C112⋊C3 in GAP, Magma, Sage, TeX

C_{11}^2\rtimes C_3
% in TeX

G:=Group("C11^2:C3");
// GroupNames label

G:=SmallGroup(363,2);
// by ID

G=gap.SmallGroup(363,2);
# by ID

G:=PCGroup([3,-3,-11,11,2107,947]);
// Polycyclic

G:=Group<a,b,c|a^11=b^11=c^3=1,a*b=b*a,c*a*c^-1=a^3*b,c*b*c^-1=a^9*b^7>;
// generators/relations

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