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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 [×2], C2 [×4], C4 [×4], C4 [×6], C22 [×3], C22 [×6], C5, C8 [×2], C2×C4 [×6], C2×C4 [×11], D4 [×7], Q8 [×3], C23, C23, D5 [×2], C10, C10 [×2], C10 [×2], C42 [×2], C42, C2×C8, M4(2) [×3], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×Q8, C4○D4 [×6], Dic5 [×2], C20 [×4], C20 [×4], D10 [×4], C2×C10 [×3], C2×C10 [×2], C4≀C2 [×4], C2×C42, C2×M4(2), C2×C4○D4, C52C8 [×2], Dic10 [×2], Dic10, C4×D5 [×4], D20 [×2], D20, C2×Dic5, C5⋊D4 [×4], C2×C20 [×6], C2×C20 [×6], C22×D5, C22×C10, C2×C4≀C2, C2×C52C8, C4.Dic5 [×2], C4.Dic5, C4×C20 [×2], C4×C20, C2×Dic10, C2×C4×D5, C2×D20, C4○D20 [×4], C4○D20 [×2], C2×C5⋊D4, C22×C20, C22×C20, D204C4 [×4], C2×C4.Dic5, C2×C4×C20, C2×C4○D20, C2×D204C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C4≀C2 [×2], C2×C22⋊C4, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C2×C4≀C2, D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, D204C4 [×2], 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 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 77)(2 76)(3 75)(4 74)(5 73)(6 72)(7 71)(8 70)(9 69)(10 68)(11 67)(12 66)(13 65)(14 64)(15 63)(16 62)(17 61)(18 80)(19 79)(20 78)(21 48)(22 47)(23 46)(24 45)(25 44)(26 43)(27 42)(28 41)(29 60)(30 59)(31 58)(32 57)(33 56)(34 55)(35 54)(36 53)(37 52)(38 51)(39 50)(40 49)
(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 70 51 80)(42 71 52 61)(43 72 53 62)(44 73 54 63)(45 74 55 64)(46 75 56 65)(47 76 57 66)(48 77 58 67)(49 78 59 68)(50 79 60 69)

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

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

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

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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