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G = C2×C11⋊C20  order 440 = 23·5·11

Direct product of C2 and C11⋊C20

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

Aliases: C2×C11⋊C20, C22⋊C20, C22.F11, Dic11⋊3C10, C11⋊2(C2×C20), (C2×C22).C10, (C2×Dic11)⋊C5, C2.2(C2×F11), C22.4(C2×C10), (C2×C11⋊C5)⋊C4, C11⋊C5⋊2(C2×C4), (C22×C11⋊C5).C2, (C2×C11⋊C5).4C22, SmallGroup(440,10)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C11 — C2×C11⋊C20
Chief series
C1 — C11 — C22 — C2×C11⋊C5 — C11⋊C20 — C2×C11⋊C20
Lower central
C11 — C2×C11⋊C20
Upper central
C1 — C22

Generators and relations for C2×C11⋊C20
 G = < a,b,c | a2=b11=c20=1, ab=ba, ac=ca, cbc-1=b2 >

C1
C2
C2
C2
11C5
C11
C22
11C4
11C4
11C10
11C10
11C10
C22
C22
C22
C11⋊C5
11C2×C4
11C20
11C20
11C2×C10
Dic11
C2×C22
Dic11
C2×C11⋊C5
C2×C11⋊C5
C2×C11⋊C5
11C2×C20
C2×Dic11
C11⋊C20
C11⋊C20
C22×C11⋊C5
C2×C11⋊C20

Smallest permutation representation of C2×C11⋊C20
►On 88 points
Generators in S88
(1 5)(2 6)(3 7)(4 8)(9 79)(10 80)(11 81)(12 82)(13 83)(14 84)(15 85)(16 86)(17 87)(18 88)(19 69)(20 70)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 52)(30 53)(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)(37 60)(38 61)(39 62)(40 63)(41 64)(42 65)(43 66)(44 67)(45 68)(46 49)(47 50)(48 51)

(1 77 61 69 85 73 53 65 49 81 57)(2 62 86 54 50 58 78 70 74 66 82)(3 87 51 79 75 83 63 55 59 71 67)(4 52 76 64 60 68 88 80 84 56 72)(5 27 38 19 15 23 30 42 46 11 34)(6 39 16 31 47 35 28 20 24 43 12)(7 17 48 9 25 13 40 32 36 21 44)(8 29 26 41 37 45 18 10 14 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 56 57 58 59 60 61 62 63 64 65 66 67 68)(69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88)
 
G:=sub<Sym(88)| (1,5)(2,6)(3,7)(4,8)(9,79)(10,80)(11,81)(12,82)(13,83)(14,84)(15,85)(16,86)(17,87)(18,88)(19,69)(20,70)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,52)(30,53)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63)(41,64)(42,65)(43,66)(44,67)(45,68)(46,49)(47,50)(48,51), (1,77,61,69,85,73,53,65,49,81,57)(2,62,86,54,50,58,78,70,74,66,82)(3,87,51,79,75,83,63,55,59,71,67)(4,52,76,64,60,68,88,80,84,56,72)(5,27,38,19,15,23,30,42,46,11,34)(6,39,16,31,47,35,28,20,24,43,12)(7,17,48,9,25,13,40,32,36,21,44)(8,29,26,41,37,45,18,10,14,33,22), (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88)>;
 
G:=Group( (1,5)(2,6)(3,7)(4,8)(9,79)(10,80)(11,81)(12,82)(13,83)(14,84)(15,85)(16,86)(17,87)(18,88)(19,69)(20,70)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,52)(30,53)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63)(41,64)(42,65)(43,66)(44,67)(45,68)(46,49)(47,50)(48,51), (1,77,61,69,85,73,53,65,49,81,57)(2,62,86,54,50,58,78,70,74,66,82)(3,87,51,79,75,83,63,55,59,71,67)(4,52,76,64,60,68,88,80,84,56,72)(5,27,38,19,15,23,30,42,46,11,34)(6,39,16,31,47,35,28,20,24,43,12)(7,17,48,9,25,13,40,32,36,21,44)(8,29,26,41,37,45,18,10,14,33,22), (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88) );
 
G=PermutationGroup([[(1,5),(2,6),(3,7),(4,8),(9,79),(10,80),(11,81),(12,82),(13,83),(14,84),(15,85),(16,86),(17,87),(18,88),(19,69),(20,70),(21,71),(22,72),(23,73),(24,74),(25,75),(26,76),(27,77),(28,78),(29,52),(30,53),(31,54),(32,55),(33,56),(34,57),(35,58),(36,59),(37,60),(38,61),(39,62),(40,63),(41,64),(42,65),(43,66),(44,67),(45,68),(46,49),(47,50),(48,51)], [(1,77,61,69,85,73,53,65,49,81,57),(2,62,86,54,50,58,78,70,74,66,82),(3,87,51,79,75,83,63,55,59,71,67),(4,52,76,64,60,68,88,80,84,56,72),(5,27,38,19,15,23,30,42,46,11,34),(6,39,16,31,47,35,28,20,24,43,12),(7,17,48,9,25,13,40,32,36,21,44),(8,29,26,41,37,45,18,10,14,33,22)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68),(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88)]])
 

44 conjugacy classes

class 1 2A2B2C4A4B4C4D5A5B5C5D10A···10L 11 20A···20P22A22B22C
order12224444555510···101120···20222222
size1111111111111111111111···111011···11101010

44 irreducible representations

dim11111111101010
type++++-+
imageC1C2C2C4C5C10C10C20F11C11⋊C20C2×F11
kernelC2×C11⋊C20C11⋊C20C22×C11⋊C5C2×C11⋊C5C2×Dic11Dic11C2×C22C22C22C2C2
# reps121448416121

Matrix representation of C2×C11⋊C20 ►in GL12(𝔽661)

66000000000000
06600000000000
001000000000
000100000000
000010000000
000001000000
000000100000
000000010000
000000001000
000000000100
000000000010
000000000001
,
100000000000
010000000000
00000000000660
00100000000660
00010000000660
00001000000660
00000100000660
00000010000660
00000001000660
00000000100660
00000000010660
00000000001660
,
35100000000000
04710000000000
00530240530013113105300
005301315305540013113100
00530053005302413100131
00013153053053013102400
00530131005300013153024
005541315300013100530131
000055405301311311315300
005300053055413101310131
00005305300024131530131
000131053053001310554131

G:=sub<GL(12,GF(661))| [660,0,0,0,0,0,0,0,0,0,0,0,0,660,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,660,660,660,660,660,660,660,660,660,660],[351,0,0,0,0,0,0,0,0,0,0,0,0,471,0,0,0,0,0,0,0,0,0,0,0,0,530,530,530,0,530,554,0,530,0,0,0,0,24,131,0,131,131,131,0,0,0,131,0,0,0,530,530,530,0,530,554,0,530,0,0,0,530,554,0,530,0,0,0,530,530,530,0,0,0,0,530,530,530,0,530,554,0,530,0,0,131,0,24,131,0,131,131,131,0,0,0,0,131,131,131,0,0,0,131,0,24,131,0,0,0,131,0,24,131,0,131,131,131,0,0,0,530,0,0,0,530,530,530,0,530,554,0,0,0,0,131,0,24,131,0,131,131,131] >;
 

C2×C11⋊C20 in GAP, Magma, Sage, TeX 

C_2\times C_{11}\rtimes C_{20}
 
% in TeX
 
G:=Group("C2xC11:C20");
 
// GroupNames label
 
G:=SmallGroup(440,10);
 
// by ID
 
G=gap.SmallGroup(440,10);
 
# by ID
 
G:=PCGroup([5,-2,-2,-5,-2,-11,100,10004,2264]);
 
// Polycyclic
 
G:=Group<a,b,c|a^2=b^11=c^20=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^2>;
 
// generators/relations
 

Export

Subgroup lattice of C2×C11⋊C20 in TeX

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