Copied to
clipboard

## G = C2×C11⋊C20order 440 = 23·5·11

### Direct product of C2 and C11⋊C20

Aliases: C2×C11⋊C20, C22⋊C20, C22.F11, Dic113C10, C112(C2×C20), (C2×C22).C10, (C2×Dic11)⋊C5, C2.2(C2×F11), C22.4(C2×C10), (C2×C11⋊C5)⋊C4, C11⋊C52(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 >

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 2A 2B 2C 4A 4B 4C 4D 5A 5B 5C 5D 10A ··· 10L 11 20A ··· 20P 22A 22B 22C order 1 2 2 2 4 4 4 4 5 5 5 5 10 ··· 10 11 20 ··· 20 22 22 22 size 1 1 1 1 11 11 11 11 11 11 11 11 11 ··· 11 10 11 ··· 11 10 10 10

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 10 10 10 type + + + + - + image C1 C2 C2 C4 C5 C10 C10 C20 F11 C11⋊C20 C2×F11 kernel C2×C11⋊C20 C11⋊C20 C22×C11⋊C5 C2×C11⋊C5 C2×Dic11 Dic11 C2×C22 C22 C22 C2 C2 # reps 1 2 1 4 4 8 4 16 1 2 1

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

 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 0 0 0 0 0 0 0 0 660 0 0 1 0 0 0 0 0 0 0 0 660 0 0 0 1 0 0 0 0 0 0 0 660 0 0 0 0 1 0 0 0 0 0 0 660 0 0 0 0 0 1 0 0 0 0 0 660 0 0 0 0 0 0 1 0 0 0 0 660 0 0 0 0 0 0 0 1 0 0 0 660 0 0 0 0 0 0 0 0 1 0 0 660 0 0 0 0 0 0 0 0 0 1 0 660 0 0 0 0 0 0 0 0 0 0 1 660
,
 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 24 0 530 0 131 131 0 530 0 0 0 530 131 530 554 0 0 131 131 0 0 0 0 530 0 530 0 530 24 131 0 0 131 0 0 0 131 530 530 530 131 0 24 0 0 0 0 530 131 0 0 530 0 0 131 530 24 0 0 554 131 530 0 0 131 0 0 530 131 0 0 0 0 554 0 530 131 131 131 530 0 0 0 530 0 0 530 554 131 0 131 0 131 0 0 0 0 530 530 0 0 24 131 530 131 0 0 0 131 0 530 530 0 131 0 554 131

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

׿
×
𝔽