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## G = C2×D20⋊4C4order 320 = 26·5

### Direct product of C2 and D20⋊4C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×D20⋊4C4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C4○D20 — C2×C4○D20 — C2×D20⋊4C4
 Lower central C5 — C10 — C20 — C2×D20⋊4C4
 Upper central C1 — C2×C4 — C22×C4 — C2×C42

Generators and relations for C2×D204C4
G = < a,b,c,d | a2=b20=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b15c >

Subgroups: 606 in 170 conjugacy classes, 63 normal (41 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, C42, C42, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C4≀C2, C2×C42, C2×M4(2), C2×C4○D4, C52C8, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C2×C4≀C2, C2×C52C8, C4.Dic5, C4.Dic5, C4×C20, C4×C20, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C4○D20, C2×C5⋊D4, C22×C20, C22×C20, D204C4, C2×C4.Dic5, C2×C4×C20, C2×C4○D20, C2×D204C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C4≀C2, C2×C22⋊C4, C4×D5, D20, C5⋊D4, C22×D5, C2×C4≀C2, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, D204C4, C2×D10⋊C4, C2×D204C4

Smallest permutation representation of C2×D204C4
On 80 points
Generators in S80
(1 36)(2 37)(3 38)(4 39)(5 40)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(41 68)(42 69)(43 70)(44 71)(45 72)(46 73)(47 74)(48 75)(49 76)(50 77)(51 78)(52 79)(53 80)(54 61)(55 62)(56 63)(57 64)(58 65)(59 66)(60 67)
(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 63)(2 62)(3 61)(4 80)(5 79)(6 78)(7 77)(8 76)(9 75)(10 74)(11 73)(12 72)(13 71)(14 70)(15 69)(16 68)(17 67)(18 66)(19 65)(20 64)(21 51)(22 50)(23 49)(24 48)(25 47)(26 46)(27 45)(28 44)(29 43)(30 42)(31 41)(32 60)(33 59)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)
(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 73 51 63)(42 74 52 64)(43 75 53 65)(44 76 54 66)(45 77 55 67)(46 78 56 68)(47 79 57 69)(48 80 58 70)(49 61 59 71)(50 62 60 72)

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

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

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

92 conjugacy classes

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

92 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 D4 D5 D10 D10 C4≀C2 C4×D5 D20 C5⋊D4 C5⋊D4 D20⋊4C4 kernel C2×D20⋊4C4 D20⋊4C4 C2×C4.Dic5 C2×C4×C20 C2×C4○D20 C2×Dic10 C2×D20 C4○D20 C2×C20 C22×C10 C2×C42 C42 C22×C4 C10 C2×C4 C2×C4 C2×C4 C23 C2 # reps 1 4 1 1 1 2 2 4 3 1 2 4 2 8 8 8 4 4 32

Matrix representation of C2×D204C4 in GL3(𝔽41) generated by

 40 0 0 0 40 0 0 0 40
,
 1 0 0 0 5 0 0 0 33
,
 1 0 0 0 0 8 0 36 0
,
 9 0 0 0 1 0 0 0 9
G:=sub<GL(3,GF(41))| [40,0,0,0,40,0,0,0,40],[1,0,0,0,5,0,0,0,33],[1,0,0,0,0,36,0,8,0],[9,0,0,0,1,0,0,0,9] >;

C2×D204C4 in GAP, Magma, Sage, TeX

C_2\times D_{20}\rtimes_4C_4
% in TeX

G:=Group("C2xD20:4C4");
// GroupNames label

G:=SmallGroup(320,554);
// by ID

G=gap.SmallGroup(320,554);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,422,58,1123,1684,102,12550]);
// 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,b*d=d*b,d*c*d^-1=b^15*c>;
// generators/relations

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