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## G = C2×D20⋊C4order 320 = 26·5

### Direct product of C2 and D20⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×D20⋊C4
 Chief series C1 — C5 — C10 — D10 — C4×D5 — C4⋊F5 — C2×C4⋊F5 — C2×D20⋊C4
 Lower central C5 — C10 — C20 — C2×D20⋊C4
 Upper central C1 — C22 — C2×C4 — C2×D4

Generators and relations for C2×D20⋊C4
G = < a,b,c,d | a2=b20=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b17c >

Subgroups: 1066 in 202 conjugacy classes, 60 normal (32 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×4], C22, C22 [×22], C5, C8 [×2], C2×C4, C2×C4 [×9], D4 [×2], D4 [×8], C23 [×11], D5 [×2], D5 [×2], D5 [×2], C10, C10 [×2], C10 [×2], C4⋊C4 [×3], C2×C8 [×4], C22×C4 [×2], C2×D4, C2×D4 [×8], C24, Dic5 [×2], C20 [×2], F5 [×2], D10 [×2], D10 [×4], D10 [×12], C2×C10, C2×C10 [×4], D4⋊C4 [×4], C2×C4⋊C4, C22×C8, C22×D4, C5⋊C8 [×2], C4×D5 [×4], D20 [×2], D20, C2×Dic5, C5⋊D4 [×4], C2×C20, C5×D4 [×2], C5×D4, C2×F5 [×4], C22×D5, C22×D5 [×9], C22×C10, C2×D4⋊C4, D5⋊C8 [×2], D5⋊C8, C4⋊F5 [×2], C4⋊F5, C2×C5⋊C8, C2×C4×D5, C2×D20, D4×D5 [×4], D4×D5 [×2], C2×C5⋊D4, D4×C10, C22×F5, C23×D5, D20⋊C4 [×4], C2×D5⋊C8, C2×C4⋊F5, C2×D4×D5, C2×D20⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], F5, D4⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16, C2×F5 [×3], C2×D4⋊C4, C22⋊F5 [×2], C22×F5, D20⋊C4 [×2], C2×C22⋊F5, C2×D20⋊C4

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 4 8 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 D4 D4 D8 SD16 F5 C2×F5 C2×F5 C22⋊F5 C22⋊F5 D20⋊C4 kernel C2×D20⋊C4 D20⋊C4 C2×D5⋊C8 C2×C4⋊F5 C2×D4×D5 C2×D20 D4×D5 D4×C10 C4×D5 C2×Dic5 C22×D5 D10 D10 C2×D4 C2×C4 D4 C4 C22 C2 # reps 1 4 1 1 1 2 4 2 2 1 1 4 4 1 1 2 2 2 2

Matrix representation of C2×D20⋊C4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 5 0 0 0 0 16 1 0 0 0 0 0 0 0 40 0 0 0 0 1 35 0 0 0 0 0 0 1 35 0 0 0 0 6 6
,
 1 36 0 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 40 0 0 0 0 0 0 0 40 6 0 0 0 0 0 1
,
 17 19 0 0 0 0 28 24 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 1 0 0 0 0 0 6 40 0 0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,16,0,0,0,0,5,1,0,0,0,0,0,0,0,1,0,0,0,0,40,35,0,0,0,0,0,0,1,6,0,0,0,0,35,6],[1,0,0,0,0,0,36,40,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,6,1],[17,28,0,0,0,0,19,24,0,0,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,40,0,0,0,0,0,0,40,0,0] >;

C2×D20⋊C4 in GAP, Magma, Sage, TeX

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

G:=Group("C2xD20:C4");
// GroupNames label

G:=SmallGroup(320,1104);
// by ID

G=gap.SmallGroup(320,1104);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,1684,438,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^17*c>;
// generators/relations

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