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G = C22×F11order 440 = 23·5·11

Direct product of C22 and F11

Aliases: C22×F11, D223C10, C11⋊C5⋊C23, C22⋊(C2×C10), D11⋊(C2×C10), C11⋊(C22×C10), (C22×D11)⋊C5, (C2×C22)⋊2C10, (C2×C11⋊C5)⋊C22, (C22×C11⋊C5)⋊2C2, SmallGroup(440,42)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C11 — C22×F11
 Chief series C1 — C11 — C11⋊C5 — F11 — C2×F11 — C22×F11
 Lower central C11 — C22×F11
 Upper central C1 — C22

Generators and relations for C22×F11
G = < a,b,c,d | a2=b2=c11=d10=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c6 >

Subgroups: 334 in 64 conjugacy classes, 37 normal (8 characteristic)
C1, C2 [×3], C2 [×4], C22, C22 [×6], C5, C23, C10 [×7], C11, C2×C10 [×7], D11 [×4], C22 [×3], C22×C10, D22 [×6], C2×C22, C11⋊C5, C22×D11, F11 [×4], C2×C11⋊C5 [×3], C2×F11 [×6], C22×C11⋊C5, C22×F11
Quotients: C1, C2 [×7], C22 [×7], C5, C23, C10 [×7], C2×C10 [×7], C22×C10, F11, C2×F11 [×3], C22×F11

Smallest permutation representation of C22×F11
On 44 points
Generators in S44
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(23 34)(24 35)(25 36)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)(33 44)
(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)
(1 34)(2 36 5 42 6 44 10 41 4 40)(3 38 9 39 11 43 8 37 7 35)(12 23)(13 25 16 31 17 33 21 30 15 29)(14 27 20 28 22 32 19 26 18 24)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 5A 5B 5C 5D 10A ··· 10AB 11 22A 22B 22C order 1 2 2 2 2 2 2 2 5 5 5 5 10 ··· 10 11 22 22 22 size 1 1 1 1 11 11 11 11 11 11 11 11 11 ··· 11 10 10 10 10

44 irreducible representations

 dim 1 1 1 1 1 1 10 10 type + + + + + image C1 C2 C2 C5 C10 C10 F11 C2×F11 kernel C22×F11 C2×F11 C22×C11⋊C5 C22×D11 D22 C2×C22 C22 C2 # reps 1 6 1 4 24 4 1 3

Matrix representation of C22×F11 in GL11(𝔽331)

 330 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1
,
 330 0 0 0 0 0 0 0 0 0 0 0 330 0 0 0 0 0 0 0 0 0 0 0 330 0 0 0 0 0 0 0 0 0 0 0 330 0 0 0 0 0 0 0 0 0 0 0 330 0 0 0 0 0 0 0 0 0 0 0 330 0 0 0 0 0 0 0 0 0 0 0 330 0 0 0 0 0 0 0 0 0 0 0 330 0 0 0 0 0 0 0 0 0 0 0 330 0 0 0 0 0 0 0 0 0 0 0 330 0 0 0 0 0 0 0 0 0 0 0 330
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 330 0 1 0 0 0 0 0 0 0 0 330 0 0 1 0 0 0 0 0 0 0 330 0 0 0 1 0 0 0 0 0 0 330 0 0 0 0 1 0 0 0 0 0 330 0 0 0 0 0 1 0 0 0 0 330 0 0 0 0 0 0 1 0 0 0 330 0 0 0 0 0 0 0 1 0 0 330 0 0 0 0 0 0 0 0 1 0 330 0 0 0 0 0 0 0 0 0 1 330
,
 207 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 330 0 0 0 0 0 330 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 330 0 0 0 0 0 330 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 330 0 0 0 0 0 330 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 330 0 0 0 0 0 330 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 330 0 0 0 0 0 330 0 0 0 0 0

G:=sub<GL(11,GF(331))| [330,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1],[330,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,330],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,330,330,330,330,330,330,330,330,330,330],[207,0,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,0,330,0,330,0,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,0,330,0] >;

C22×F11 in GAP, Magma, Sage, TeX

C_2^2\times F_{11}
% in TeX

G:=Group("C2^2xF11");
// GroupNames label

G:=SmallGroup(440,42);
// by ID

G=gap.SmallGroup(440,42);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-11,10004,1144]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^11=d^10=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^6>;
// generators/relations

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