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## G = (C2×C20).D4order 320 = 26·5

### 2nd non-split extension by C2×C20 of D4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C10 — (C2×C20).D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — D4×C10 — C23.18D10 — (C2×C20).D4
 Lower central C5 — C10 — C2×C10 — C22×C10 — (C2×C20).D4
 Upper central C1 — C2 — C22 — C2×D4 — C23⋊C4

Generators and relations for (C2×C20).D4
G = < a,b,c,d | a2=b4=c20=1, d2=b-1, ab=ba, cac-1=dad-1=ab2, cbc-1=ab-1, bd=db, dcd-1=bc-1 >

Subgroups: 286 in 68 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C10, C10, C22⋊C4, C4⋊C4, M4(2), C22×C4, C2×D4, Dic5, C20, C2×C10, C2×C10, C23⋊C4, C4.D4, C22.D4, C52C8, C2×Dic5, C2×C20, C2×C20, C5×D4, C22×C10, C23.D4, C4.Dic5, C10.D4, C23.D5, C23.D5, C5×C22⋊C4, C22×Dic5, D4×C10, C20.D4, C5×C23⋊C4, C23.18D10, (C2×C20).D4
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, D10, C23⋊C4, C4×D5, D20, C5⋊D4, C23.D4, D10⋊C4, C23.1D10, (C2×C20).D4

Smallest permutation representation of (C2×C20).D4
On 80 points
Generators in S80
(1 70)(3 72)(5 74)(7 76)(9 78)(11 80)(13 62)(15 64)(17 66)(19 68)(22 47)(24 49)(26 51)(28 53)(30 55)(32 57)(34 59)(36 41)(38 43)(40 45)
(1 28 70 53)(2 29 71 54)(3 55 72 30)(4 56 73 31)(5 32 74 57)(6 33 75 58)(7 59 76 34)(8 60 77 35)(9 36 78 41)(10 37 79 42)(11 43 80 38)(12 44 61 39)(13 40 62 45)(14 21 63 46)(15 47 64 22)(16 48 65 23)(17 24 66 49)(18 25 67 50)(19 51 68 26)(20 52 69 27)
(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 37 53 10 70 42 28 79)(2 78 54 36 71 9 29 41)(3 35 30 77 72 60 55 8)(4 7 31 34 73 76 56 59)(5 33 57 6 74 58 32 75)(11 27 38 69 80 52 43 20)(12 19 39 26 61 68 44 51)(13 25 45 18 62 50 40 67)(14 66 46 24 63 17 21 49)(15 23 22 65 64 48 47 16)

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

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

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

35 conjugacy classes

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

35 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 8 type + + + + + + + + + + - image C1 C2 C2 C2 C4 C4 D4 D4 D5 D10 D20 C4×D5 C5⋊D4 C23⋊C4 C23.D4 C23.1D10 (C2×C20).D4 kernel (C2×C20).D4 C20.D4 C5×C23⋊C4 C23.18D10 C23.D5 C22×Dic5 C2×C20 C22×C10 C23⋊C4 C2×D4 C2×C4 C23 C23 C10 C5 C2 C1 # reps 1 1 1 1 2 2 1 1 2 2 4 4 4 1 2 4 2

Matrix representation of (C2×C20).D4 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 20 20 1 0 0 0 19 19 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 0 0 0 0 0 0 32 0 0 0 0 25 0 32 0 0 0 0 7 0 9
,
 0 40 0 0 0 0 1 7 0 0 0 0 0 0 8 8 9 0 0 0 24 24 0 9 0 0 8 7 33 33 0 0 25 24 17 17
,
 0 40 0 0 0 0 40 0 0 0 0 0 0 0 11 11 0 40 0 0 31 31 40 0 0 0 31 22 10 10 0 0 20 11 30 30

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,20,19,0,0,0,40,20,19,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,25,0,0,0,0,32,0,7,0,0,0,0,32,0,0,0,0,0,0,9],[0,1,0,0,0,0,40,7,0,0,0,0,0,0,8,24,8,25,0,0,8,24,7,24,0,0,9,0,33,17,0,0,0,9,33,17],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,11,31,31,20,0,0,11,31,22,11,0,0,0,40,10,30,0,0,40,0,10,30] >;

(C2×C20).D4 in GAP, Magma, Sage, TeX

(C_2\times C_{20}).D_4
% in TeX

G:=Group("(C2xC20).D4");
// GroupNames label

G:=SmallGroup(320,30);
// by ID

G=gap.SmallGroup(320,30);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,141,36,422,346,297,851,12550]);
// Polycyclic

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

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