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## 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, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, C23, C10, C10, C10, C2×C8, M4(2), C22×C4, C2×D4, C2×D4, C24, C20, C2×C10, C2×C10, C4.D4, C2×M4(2), C22×D4, C52C8, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C22×C10, C2×C4.D4, C2×C52C8, C4.Dic5, C4.Dic5, C22×C20, D4×C10, D4×C10, C23×C10, C20.D4, C2×C4.Dic5, D4×C2×C10, C2×C20.D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, Dic5, D10, C4.D4, C2×C22⋊C4, C2×Dic5, C5⋊D4, C22×D5, C2×C4.D4, C23.D5, C22×Dic5, C2×C5⋊D4, C20.D4, C2×C23.D5, C2×C20.D4

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

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

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

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

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

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