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## G = C5×C11⋊C5order 275 = 52·11

### Direct product of C5 and C11⋊C5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C11 — C5×C11⋊C5
 Chief series C1 — C11 — C11⋊C5 — C5×C11⋊C5
 Lower central C11 — C5×C11⋊C5
 Upper central C1 — C5

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

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 5A 5B 5C 5D 5E ··· 5X 11A 11B 55A ··· 55H order 1 5 5 5 5 5 ··· 5 11 11 55 ··· 55 size 1 1 1 1 1 11 ··· 11 5 5 5 ··· 5

35 irreducible representations

 dim 1 1 1 5 5 type + image C1 C5 C5 C11⋊C5 C5×C11⋊C5 kernel C5×C11⋊C5 C11⋊C5 C55 C5 C1 # reps 1 20 4 2 8

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

 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 2 103 226 1 0 226 2 103 226 1 0 225 3 103 226 1 0 225 2 104 226 1 0 225 2 103 227 1
,
 323 0 0 0 0 0 0 0 0 1 0 0 0 227 106 329 228 105 0 228 105 225 2 104 0 1 0 0 0 0 0 0 0 0 1 0

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

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