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## G = C2×C4○D20order 160 = 25·5

### Direct product of C2 and C4○D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C4○D20
 Chief series C1 — C5 — C10 — D10 — C22×D5 — C2×C4×D5 — C2×C4○D20
 Lower central C5 — C10 — C2×C4○D20
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C2×C4○D20
G = < a,b,c,d | a2=b4=d2=1, c10=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c9 >

Subgroups: 456 in 164 conjugacy classes, 89 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C22×C20, C2×C4○D20
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, C22×D5, C4○D20, C23×D5, C2×C4○D20

Smallest permutation representation of C2×C4○D20
On 80 points
Generators in S80
(1 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 40)(18 21)(19 22)(20 23)(41 79)(42 80)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)
(1 79 11 69)(2 80 12 70)(3 61 13 71)(4 62 14 72)(5 63 15 73)(6 64 16 74)(7 65 17 75)(8 66 18 76)(9 67 19 77)(10 68 20 78)(21 58 31 48)(22 59 32 49)(23 60 33 50)(24 41 34 51)(25 42 35 52)(26 43 36 53)(27 44 37 54)(28 45 38 55)(29 46 39 56)(30 47 40 57)
(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 23)(2 22)(3 21)(4 40)(5 39)(6 38)(7 37)(8 36)(9 35)(10 34)(11 33)(12 32)(13 31)(14 30)(15 29)(16 28)(17 27)(18 26)(19 25)(20 24)(41 78)(42 77)(43 76)(44 75)(45 74)(46 73)(47 72)(48 71)(49 70)(50 69)(51 68)(52 67)(53 66)(54 65)(55 64)(56 63)(57 62)(58 61)(59 80)(60 79)

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

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

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

52 conjugacy classes

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

52 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 C4○D20 kernel C2×C4○D20 C2×Dic10 C2×C4×D5 C2×D20 C4○D20 C2×C5⋊D4 C22×C20 C22×C4 C10 C2×C4 C23 C2 # reps 1 1 2 1 8 2 1 2 4 12 2 16

Matrix representation of C2×C4○D20 in GL3(𝔽41) generated by

 40 0 0 0 40 0 0 0 40
,
 1 0 0 0 32 0 0 0 32
,
 40 0 0 0 39 27 0 30 25
,
 40 0 0 0 39 4 0 30 2
G:=sub<GL(3,GF(41))| [40,0,0,0,40,0,0,0,40],[1,0,0,0,32,0,0,0,32],[40,0,0,0,39,30,0,27,25],[40,0,0,0,39,30,0,4,2] >;

C2×C4○D20 in GAP, Magma, Sage, TeX

C_2\times C_4\circ D_{20}
% in TeX

G:=Group("C2xC4oD20");
// GroupNames label

G:=SmallGroup(160,216);
// by ID

G=gap.SmallGroup(160,216);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,86,579,4613]);
// Polycyclic

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

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