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## G = C2×C5⋊2C8order 80 = 24·5

### Direct product of C2 and C5⋊2C8

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

 Derived series C1 — C5 — C2×C5⋊2C8
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C2×C5⋊2C8
 Lower central C5 — C2×C5⋊2C8
 Upper central C1 — C2×C4

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

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 2A 2B 2C 4A 4B 4C 4D 5A 5B 8A ··· 8H 10A ··· 10F 20A ··· 20H order 1 2 2 2 4 4 4 4 5 5 8 ··· 8 10 ··· 10 20 ··· 20 size 1 1 1 1 1 1 1 1 2 2 5 ··· 5 2 ··· 2 2 ··· 2

32 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C4 C4 C8 D5 Dic5 D10 Dic5 C5⋊2C8 kernel C2×C5⋊2C8 C5⋊2C8 C2×C20 C20 C2×C10 C10 C2×C4 C4 C4 C22 C2 # reps 1 2 1 2 2 8 2 2 2 2 8

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

 40 0 0 0 1 0 0 0 1
,
 1 0 0 0 34 40 0 1 0
,
 1 0 0 0 31 3 0 32 10
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

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