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G = C5×C8○D4order 160 = 25·5

Direct product of C5 and C8○D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C5×C8○D4, D4.C20, Q8.C20, M4(2)⋊5C10, C20.54C23, C40.30C22, C40(C5×D4), C40(C5×Q8), (C2×C8)⋊7C10, (C2×C40)⋊15C2, C4.5(C2×C20), C8.7(C2×C10), (C5×D4).3C4, (C5×Q8).3C4, C20.53(C2×C4), C4○D4.3C10, C40(C5×M4(2)), C2.7(C22×C20), C22.1(C2×C20), (C5×M4(2))⋊11C2, C10.48(C22×C4), C4.12(C22×C10), (C2×C20).128C22, C40(C5×C4○D4), (C5×C4○D4).6C2, (C2×C10).28(C2×C4), (C2×C4).24(C2×C10), SmallGroup(160,192)

Series: Derived Chief Lower central Upper central

C1C2 — C5×C8○D4
C1C2C4C20C40C2×C40 — C5×C8○D4
C1C2 — C5×C8○D4
C1C40 — C5×C8○D4

Generators and relations for C5×C8○D4
 G = < a,b,c,d | a5=b8=d2=1, c2=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >

Subgroups: 68 in 62 conjugacy classes, 56 normal (14 characteristic)
C1, C2, C2 [×3], C4, C4 [×3], C22 [×3], C5, C8, C8 [×3], C2×C4 [×3], D4 [×3], Q8, C10, C10 [×3], C2×C8 [×3], M4(2) [×3], C4○D4, C20, C20 [×3], C2×C10 [×3], C8○D4, C40, C40 [×3], C2×C20 [×3], C5×D4 [×3], C5×Q8, C2×C40 [×3], C5×M4(2) [×3], C5×C4○D4, C5×C8○D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], C23, C10 [×7], C22×C4, C20 [×4], C2×C10 [×7], C8○D4, C2×C20 [×6], C22×C10, C22×C20, C5×C8○D4

Smallest permutation representation of C5×C8○D4
On 80 points
Generators in S80
(1 57 71 17 31)(2 58 72 18 32)(3 59 65 19 25)(4 60 66 20 26)(5 61 67 21 27)(6 62 68 22 28)(7 63 69 23 29)(8 64 70 24 30)(9 40 44 54 80)(10 33 45 55 73)(11 34 46 56 74)(12 35 47 49 75)(13 36 48 50 76)(14 37 41 51 77)(15 38 42 52 78)(16 39 43 53 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 35 5 39)(2 36 6 40)(3 37 7 33)(4 38 8 34)(9 32 13 28)(10 25 14 29)(11 26 15 30)(12 27 16 31)(17 75 21 79)(18 76 22 80)(19 77 23 73)(20 78 24 74)(41 63 45 59)(42 64 46 60)(43 57 47 61)(44 58 48 62)(49 67 53 71)(50 68 54 72)(51 69 55 65)(52 70 56 66)
(1 39)(2 40)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 79)(18 80)(19 73)(20 74)(21 75)(22 76)(23 77)(24 78)(41 63)(42 64)(43 57)(44 58)(45 59)(46 60)(47 61)(48 62)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 65)(56 66)

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

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

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

C5×C8○D4 is a maximal subgroup of
C40.92D4  C40.70C23  D4.3D20  D4.4D20  D4.5D20  C40.93D4  C40.50D4  C20.72C24  D4.11D20  D4.12D20  D4.13D20
C5×C8○D4 is a maximal quotient of
D4×C40  Q8×C40

100 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E5A5B5C5D8A8B8C8D8E···8J10A10B10C10D10E···10P20A···20H20I···20T40A···40P40Q···40AN
order1222244444555588888···81010101010···1020···2020···2040···4040···40
size1122211222111111112···211112···21···12···21···12···2

100 irreducible representations

dim11111111111122
type++++
imageC1C2C2C2C4C4C5C10C10C10C20C20C8○D4C5×C8○D4
kernelC5×C8○D4C2×C40C5×M4(2)C5×C4○D4C5×D4C5×Q8C8○D4C2×C8M4(2)C4○D4D4Q8C5C1
# reps133162412124248416

Matrix representation of C5×C8○D4 in GL2(𝔽41) generated by

100
010
,
30
03
,
139
140
,
139
040
G:=sub<GL(2,GF(41))| [10,0,0,10],[3,0,0,3],[1,1,39,40],[1,0,39,40] >;

C5×C8○D4 in GAP, Magma, Sage, TeX

C_5\times C_8\circ D_4
% in TeX

G:=Group("C5xC8oD4");
// GroupNames label

G:=SmallGroup(160,192);
// by ID

G=gap.SmallGroup(160,192);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-2,240,764,88]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=d^2=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c>;
// generators/relations

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