Copied to
clipboard

## G = C2×C20.D4order 320 = 26·5

### Direct product of C2 and C20.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×C20.D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4.Dic5 — C2×C4.Dic5 — C2×C20.D4
 Lower central C5 — C10 — C2×C10 — C2×C20.D4
 Upper central C1 — C22 — C22×C4 — C22×D4

Generators and relations for C2×C20.D4
G = < a,b,c,d | a2=b20=1, c4=b10, d2=b5, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b9, dcd-1=b15c3 >

Subgroups: 478 in 186 conjugacy classes, 71 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×4], C22 [×3], C22 [×18], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], D4 [×8], C23, C23 [×4], C23 [×8], C10, C10 [×2], C10 [×6], C2×C8 [×2], M4(2) [×6], C22×C4, C2×D4 [×4], C2×D4 [×4], C24 [×2], C20 [×4], C2×C10 [×3], C2×C10 [×18], C4.D4 [×4], C2×M4(2) [×2], C22×D4, C52C8 [×4], C2×C20 [×2], C2×C20 [×4], C5×D4 [×8], C22×C10, C22×C10 [×4], C22×C10 [×8], C2×C4.D4, C2×C52C8 [×2], C4.Dic5 [×4], C4.Dic5 [×2], C22×C20, D4×C10 [×4], D4×C10 [×4], C23×C10 [×2], C20.D4 [×4], C2×C4.Dic5 [×2], D4×C2×C10, C2×C20.D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], Dic5 [×4], D10 [×3], C4.D4 [×2], C2×C22⋊C4, C2×Dic5 [×6], C5⋊D4 [×4], C22×D5, C2×C4.D4, C23.D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], C20.D4 [×2], C2×C23.D5, C2×C20.D4

Smallest permutation representation of C2×C20.D4
On 80 points
Generators in S80
(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 21)(41 76)(42 77)(43 78)(44 79)(45 80)(46 61)(47 62)(48 63)(49 64)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)(57 72)(58 73)(59 74)(60 75)
(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)
(1 66 6 61 11 76 16 71)(2 65 7 80 12 75 17 70)(3 64 8 79 13 74 18 69)(4 63 9 78 14 73 19 68)(5 62 10 77 15 72 20 67)(21 52 26 47 31 42 36 57)(22 51 27 46 32 41 37 56)(23 50 28 45 33 60 38 55)(24 49 29 44 34 59 39 54)(25 48 30 43 35 58 40 53)
(1 61 6 66 11 71 16 76)(2 70 7 75 12 80 17 65)(3 79 8 64 13 69 18 74)(4 68 9 73 14 78 19 63)(5 77 10 62 15 67 20 72)(21 57 26 42 31 47 36 52)(22 46 27 51 32 56 37 41)(23 55 28 60 33 45 38 50)(24 44 29 49 34 54 39 59)(25 53 30 58 35 43 40 48)

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 5A 5B 8A ··· 8H 10A ··· 10N 10O ··· 10AD 20A ··· 20H order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 5 5 8 ··· 8 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 4 4 4 4 2 2 2 2 2 2 20 ··· 20 2 ··· 2 4 ··· 4 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + - + - + image C1 C2 C2 C2 C4 C4 D4 D5 D10 Dic5 D10 Dic5 C5⋊D4 C4.D4 C20.D4 kernel C2×C20.D4 C20.D4 C2×C4.Dic5 D4×C2×C10 D4×C10 C23×C10 C2×C20 C22×D4 C22×C4 C2×D4 C2×D4 C24 C2×C4 C10 C2 # reps 1 4 2 1 4 4 4 2 2 4 4 4 16 2 8

Matrix representation of C2×C20.D4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 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
,
 38 34 0 0 0 0 34 38 0 0 0 0 0 0 0 40 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 40 0
,
 23 35 0 0 0 0 6 18 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 40 0 0 0 0 1 0 0 0
,
 35 23 0 0 0 0 18 6 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 40 0 0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[38,34,0,0,0,0,34,38,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[23,6,0,0,0,0,35,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,1,0,0],[35,18,0,0,0,0,23,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,1,0,0,0] >;

C2×C20.D4 in GAP, Magma, Sage, TeX

C_2\times C_{20}.D_4
% in TeX

G:=Group("C2xC20.D4");
// GroupNames label

G:=SmallGroup(320,843);
// by ID

G=gap.SmallGroup(320,843);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,297,136,1684,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=1,c^4=b^10,d^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^9,d*c*d^-1=b^15*c^3>;
// generators/relations

׿
×
𝔽