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G = C2×C52C8order 80 = 24·5

Direct product of C2 and C52C8

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C2×C52C8, C102C8, C20.6C4, C4.14D10, C4.3Dic5, C20.14C22, C22.2Dic5, C54(C2×C8), C4(C52C8), (C2×C4).5D5, (C2×C10).4C4, (C2×C20).6C2, C10.13(C2×C4), C2.1(C2×Dic5), SmallGroup(80,9)

Series: Derived Chief Lower central Upper central

C1C5 — C2×C52C8
C1C5C10C20C52C8 — C2×C52C8
C5 — C2×C52C8
C1C2×C4

Generators and relations for C2×C52C8
 G = < a,b,c | a2=b5=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

5C8
5C8
5C2×C8

Smallest permutation representation of C2×C52C8
Regular action on 80 points
Generators in S80
(1 60)(2 61)(3 62)(4 63)(5 64)(6 57)(7 58)(8 59)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(41 70)(42 71)(43 72)(44 65)(45 66)(46 67)(47 68)(48 69)(49 78)(50 79)(51 80)(52 73)(53 74)(54 75)(55 76)(56 77)
(1 11 25 67 75)(2 76 68 26 12)(3 13 27 69 77)(4 78 70 28 14)(5 15 29 71 79)(6 80 72 30 16)(7 9 31 65 73)(8 74 66 32 10)(17 46 54 60 38)(18 39 61 55 47)(19 48 56 62 40)(20 33 63 49 41)(21 42 50 64 34)(22 35 57 51 43)(23 44 52 58 36)(24 37 59 53 45)
(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)

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

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

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

C2×C52C8 is a maximal subgroup of
C42.D5  C203C8  C10.D8  C20.Q8  D206C4  C10.Q16  C8×Dic5  C20.8Q8  C408C4  D101C8  C20.53D4  C20.55D4  D4⋊Dic5  Q8⋊Dic5  C20.C8  D5×C2×C8  D20.2C4  D4.Dic5  D4.8D10  C20.14F5
C2×C52C8 is a maximal quotient of
C203C8  C20.4C8  C20.55D4  C20.14F5

32 conjugacy classes

class 1 2A2B2C4A4B4C4D5A5B8A···8H10A···10F20A···20H
order12224444558···810···1020···20
size11111111225···52···22···2

32 irreducible representations

dim11111122222
type++++-+-
imageC1C2C2C4C4C8D5Dic5D10Dic5C52C8
kernelC2×C52C8C52C8C2×C20C20C2×C10C10C2×C4C4C4C22C2
# reps12122822228

Matrix representation of C2×C52C8 in GL3(𝔽41) generated by

4000
010
001
,
100
03440
010
,
100
0313
03210
G:=sub<GL(3,GF(41))| [40,0,0,0,1,0,0,0,1],[1,0,0,0,34,1,0,40,0],[1,0,0,0,31,32,0,3,10] >;

C2×C52C8 in GAP, Magma, Sage, TeX

C_2\times C_5\rtimes_2C_8
% in TeX

G:=Group("C2xC5:2C8");
// GroupNames label

G:=SmallGroup(80,9);
// by ID

G=gap.SmallGroup(80,9);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,20,42,1604]);
// Polycyclic

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

Export

Subgroup lattice of C2×C52C8 in TeX

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