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G = (C2×C20).D4order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C20).2D4, (C2×C4).2D20, (C2×D4).2D10, C23⋊C4.1D5, C23.D52C4, C23.2(C4×D5), (C22×C10).9D4, C20.D4.1C2, (C22×Dic5)⋊1C4, (D4×C10).2C22, C23.2(C5⋊D4), C53(C23.D4), C10.29(C23⋊C4), C22.9(D10⋊C4), C23.18D10.1C2, C2.9(C23.1D10), (C5×C23⋊C4).1C2, (C22×C10).2(C2×C4), (C2×C10).66(C22⋊C4), SmallGroup(320,30)

Series: Derived Chief Lower central Upper central

C1C22×C10 — (C2×C20).D4
C1C5C10C2×C10C22×C10D4×C10C23.18D10 — (C2×C20).D4
C5C10C2×C10C22×C10 — (C2×C20).D4
C1C2C22C2×D4C23⋊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 [×3], C4 [×4], C22, C22 [×4], C5, C8, C2×C4, C2×C4 [×4], D4, C23 [×2], C10, C10 [×3], C22⋊C4 [×3], C4⋊C4, M4(2), C22×C4, C2×D4, Dic5 [×2], C20 [×2], C2×C10, C2×C10 [×4], C23⋊C4, C4.D4, C22.D4, C52C8, C2×Dic5 [×3], C2×C20, C2×C20, C5×D4, C22×C10 [×2], 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 [×3], C4 [×2], C22, C2×C4, D4 [×2], 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 22)(3 24)(5 26)(7 28)(9 30)(11 32)(13 34)(15 36)(17 38)(19 40)(41 74)(43 76)(45 78)(47 80)(49 62)(51 64)(53 66)(55 68)(57 70)(59 72)
(1 66 22 53)(2 67 23 54)(3 55 24 68)(4 56 25 69)(5 70 26 57)(6 71 27 58)(7 59 28 72)(8 60 29 73)(9 74 30 41)(10 75 31 42)(11 43 32 76)(12 44 33 77)(13 78 34 45)(14 79 35 46)(15 47 36 80)(16 48 37 61)(17 62 38 49)(18 63 39 50)(19 51 40 64)(20 52 21 65)
(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 75 53 10 22 42 66 31)(2 30 54 74 23 9 67 41)(3 73 68 29 24 60 55 8)(4 7 69 72 25 28 56 59)(5 71 57 6 26 58 70 27)(11 65 76 21 32 52 43 20)(12 19 77 64 33 40 44 51)(13 63 45 18 34 50 78 39)(14 38 46 62 35 17 79 49)(15 61 80 37 36 48 47 16)

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

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

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

35 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F5A5B8A8B10A10B10C···10H10I10J20A···20J
order122224444445588101010···10101020···20
size11244488202040224040224···4888···8

35 irreducible representations

dim11111122222224448
type++++++++++-
imageC1C2C2C2C4C4D4D4D5D10D20C4×D5C5⋊D4C23⋊C4C23.D4C23.1D10(C2×C20).D4
kernel(C2×C20).D4C20.D4C5×C23⋊C4C23.18D10C23.D5C22×Dic5C2×C20C22×C10C23⋊C4C2×D4C2×C4C23C23C10C5C2C1
# reps11112211224441242

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

100000
010000
0040000
0004000
00202010
00191901
,
100000
010000
009000
0003200
00250320
000709
,
0400000
170000
008890
00242409
00873333
0025241717
,
0400000
4000000
001111040
003131400
0031221010
0020113030

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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