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## G = D20.8D4order 320 = 26·5

### 8th non-split extension by D20 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D20.8D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C4×D5 — C2×D4×D5 — D20.8D4
 Lower central C5 — C10 — C2×C20 — D20.8D4
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for D20.8D4
G = < a,b,c,d | a4=b2=c20=1, d2=a2, bab=cac-1=dad-1=a-1, cbc-1=dbd-1=a-1b, dcd-1=a2c-1 >

Subgroups: 1022 in 188 conjugacy classes, 45 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, D4⋊C4, C22⋊Q8, C2×SD16, C22×D4, C52C8, C40, Dic10, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C22×D5, C22×D5, C22×C10, C22⋊SD16, C40⋊C2, C2×C52C8, C4⋊Dic5, D10⋊C4, D4.D5, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, D4×D5, C2×C5⋊D4, D4×C10, C23×D5, D206C4, D101C8, C5×D4⋊C4, D102Q8, C2×C40⋊C2, C2×D4.D5, C2×D4×D5, D20.8D4
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, D10, C22≀C2, C2×SD16, C8⋊C22, D20, C22×D5, C22⋊SD16, C2×D20, D4×D5, C22⋊D20, D8⋊D5, D5×SD16, D20.8D4

Smallest permutation representation of D20.8D4
On 80 points
Generators in S80
(1 62 39 53)(2 54 40 63)(3 64 21 55)(4 56 22 65)(5 66 23 57)(6 58 24 67)(7 68 25 59)(8 60 26 69)(9 70 27 41)(10 42 28 71)(11 72 29 43)(12 44 30 73)(13 74 31 45)(14 46 32 75)(15 76 33 47)(16 48 34 77)(17 78 35 49)(18 50 36 79)(19 80 37 51)(20 52 38 61)
(1 72)(2 30)(3 74)(4 32)(5 76)(6 34)(7 78)(8 36)(9 80)(10 38)(11 62)(12 40)(13 64)(14 22)(15 66)(16 24)(17 68)(18 26)(19 70)(20 28)(21 45)(23 47)(25 49)(27 51)(29 53)(31 55)(33 57)(35 59)(37 41)(39 43)(42 52)(44 54)(46 56)(48 58)(50 60)(61 71)(63 73)(65 75)(67 77)(69 79)
(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 38 39 20)(2 19 40 37)(3 36 21 18)(4 17 22 35)(5 34 23 16)(6 15 24 33)(7 32 25 14)(8 13 26 31)(9 30 27 12)(10 11 28 29)(41 73 70 44)(42 43 71 72)(45 69 74 60)(46 59 75 68)(47 67 76 58)(48 57 77 66)(49 65 78 56)(50 55 79 64)(51 63 80 54)(52 53 61 62)

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

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

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

47 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 5A 5B 8A 8B 8C 8D 10A ··· 10F 10G 10H 10I 10J 20A 20B 20C 20D 20E 20F 20G 20H 40A ··· 40H order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 10 10 10 10 20 20 20 20 20 20 20 20 40 ··· 40 size 1 1 1 1 4 4 10 10 20 20 2 2 8 20 40 2 2 4 4 20 20 2 ··· 2 8 8 8 8 4 4 4 4 8 8 8 8 4 ··· 4

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 D5 SD16 D10 D10 D10 D20 C8⋊C22 D4×D5 D4×D5 D8⋊D5 D5×SD16 kernel D20.8D4 D20⋊6C4 D10⋊1C8 C5×D4⋊C4 D10⋊2Q8 C2×C40⋊C2 C2×D4.D5 C2×D4×D5 D20 C2×Dic5 C5×D4 C22×D5 D4⋊C4 D10 C4⋊C4 C2×C8 C2×D4 D4 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 1 2 1 2 4 2 2 2 8 1 2 2 4 4

Matrix representation of D20.8D4 in GL6(𝔽41)

 1 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 9 0 0 0 0 18 40
,
 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 9 0 0 0 0 0 40
,
 15 37 0 0 0 0 36 26 0 0 0 0 0 0 35 40 0 0 0 0 1 0 0 0 0 0 0 0 11 29 0 0 0 0 17 30
,
 15 37 0 0 0 0 15 26 0 0 0 0 0 0 40 35 0 0 0 0 0 1 0 0 0 0 0 0 11 29 0 0 0 0 17 30

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,18,0,0,0,0,9,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,9,40],[15,36,0,0,0,0,37,26,0,0,0,0,0,0,35,1,0,0,0,0,40,0,0,0,0,0,0,0,11,17,0,0,0,0,29,30],[15,15,0,0,0,0,37,26,0,0,0,0,0,0,40,0,0,0,0,0,35,1,0,0,0,0,0,0,11,17,0,0,0,0,29,30] >;

D20.8D4 in GAP, Magma, Sage, TeX

D_{20}._8D_4
% in TeX

G:=Group("D20.8D4");
// GroupNames label

G:=SmallGroup(320,403);
// by ID

G=gap.SmallGroup(320,403);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,422,135,268,570,297,136,12550]);
// Polycyclic

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

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