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G = C5×C11⋊C5order 275 = 52·11

Direct product of C5 and C11⋊C5

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

Aliases: C5×C11⋊C5, C55⋊C5, C11⋊C52, SmallGroup(275,3)

Series: Derived Chief Lower central Upper central

C1C11 — C5×C11⋊C5
C1C11C11⋊C5 — C5×C11⋊C5
C11 — C5×C11⋊C5
C1C5

Generators and relations for C5×C11⋊C5
 G = < a,b,c | a5=b11=c5=1, ab=ba, ac=ca, cbc-1=b3 >

11C5
11C5
11C5
11C5
11C5
11C52

Smallest permutation representation of C5×C11⋊C5
On 55 points
Generators in S55
(1 45 34 23 12)(2 46 35 24 13)(3 47 36 25 14)(4 48 37 26 15)(5 49 38 27 16)(6 50 39 28 17)(7 51 40 29 18)(8 52 41 30 19)(9 53 42 31 20)(10 54 43 32 21)(11 55 44 33 22)
(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)(45 46 47 48 49 50 51 52 53 54 55)
(1 23 45 12 34)(2 27 50 21 37)(3 31 55 19 40)(4 24 49 17 43)(5 28 54 15 35)(6 32 48 13 38)(7 25 53 22 41)(8 29 47 20 44)(9 33 52 18 36)(10 26 46 16 39)(11 30 51 14 42)

G:=sub<Sym(55)| (1,45,34,23,12)(2,46,35,24,13)(3,47,36,25,14)(4,48,37,26,15)(5,49,38,27,16)(6,50,39,28,17)(7,51,40,29,18)(8,52,41,30,19)(9,53,42,31,20)(10,54,43,32,21)(11,55,44,33,22), (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)(45,46,47,48,49,50,51,52,53,54,55), (1,23,45,12,34)(2,27,50,21,37)(3,31,55,19,40)(4,24,49,17,43)(5,28,54,15,35)(6,32,48,13,38)(7,25,53,22,41)(8,29,47,20,44)(9,33,52,18,36)(10,26,46,16,39)(11,30,51,14,42)>;

G:=Group( (1,45,34,23,12)(2,46,35,24,13)(3,47,36,25,14)(4,48,37,26,15)(5,49,38,27,16)(6,50,39,28,17)(7,51,40,29,18)(8,52,41,30,19)(9,53,42,31,20)(10,54,43,32,21)(11,55,44,33,22), (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)(45,46,47,48,49,50,51,52,53,54,55), (1,23,45,12,34)(2,27,50,21,37)(3,31,55,19,40)(4,24,49,17,43)(5,28,54,15,35)(6,32,48,13,38)(7,25,53,22,41)(8,29,47,20,44)(9,33,52,18,36)(10,26,46,16,39)(11,30,51,14,42) );

G=PermutationGroup([(1,45,34,23,12),(2,46,35,24,13),(3,47,36,25,14),(4,48,37,26,15),(5,49,38,27,16),(6,50,39,28,17),(7,51,40,29,18),(8,52,41,30,19),(9,53,42,31,20),(10,54,43,32,21),(11,55,44,33,22)], [(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),(45,46,47,48,49,50,51,52,53,54,55)], [(1,23,45,12,34),(2,27,50,21,37),(3,31,55,19,40),(4,24,49,17,43),(5,28,54,15,35),(6,32,48,13,38),(7,25,53,22,41),(8,29,47,20,44),(9,33,52,18,36),(10,26,46,16,39),(11,30,51,14,42)])

35 conjugacy classes

class 1 5A5B5C5D5E···5X11A11B55A···55H
order155555···5111155···55
size1111111···11555···5

35 irreducible representations

dim11155
type+
imageC1C5C5C11⋊C5C5×C11⋊C5
kernelC5×C11⋊C5C11⋊C5C55C5C1
# reps120428

Matrix representation of C5×C11⋊C5 in GL6(𝔽331)

12400000
010000
001000
000100
000010
000001
,
100000
022521032261
022621032261
022531032261
022521042261
022521032271
,
32300000
000100
0227106329228105
02281052252104
010000
000010

G:=sub<GL(6,GF(331))| [124,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,225,226,225,225,225,0,2,2,3,2,2,0,103,103,103,104,103,0,226,226,226,226,227,0,1,1,1,1,1],[323,0,0,0,0,0,0,0,227,228,1,0,0,0,106,105,0,0,0,1,329,225,0,0,0,0,228,2,0,1,0,0,105,104,0,0] >;

C5×C11⋊C5 in GAP, Magma, Sage, TeX

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

G:=Group("C5xC11:C5");
// GroupNames label

G:=SmallGroup(275,3);
// by ID

G=gap.SmallGroup(275,3);
# by ID

G:=PCGroup([3,-5,-5,-11,902]);
// Polycyclic

G:=Group<a,b,c|a^5=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 C5×C11⋊C5 in TeX

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