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G = C2×C4×C8order 64 = 26

Abelian group of type [2,4,8]

direct product, p-group, abelian, monomial

Aliases: C2×C4×C8, SmallGroup(64,83)

Series: Derived Chief Lower central Upper central Jennings

C1 — C2×C4×C8
C1C2C22C2×C4C22×C4C2×C42 — C2×C4×C8
C1 — C2×C4×C8
C1 — C2×C4×C8
C1C2C2C2×C4 — C2×C4×C8

Generators and relations for C2×C4×C8
 G = < a,b,c | a2=b4=c8=1, ab=ba, ac=ca, bc=cb >

Subgroups: 81, all normal (9 characteristic)
C1, C2, C2 [×6], C4 [×12], C22, C22 [×6], C8 [×8], C2×C4 [×2], C2×C4 [×16], C23, C42 [×4], C2×C8 [×12], C22×C4, C22×C4 [×2], C4×C8 [×4], C2×C42, C22×C8 [×2], C2×C4×C8
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C8 [×8], C2×C4 [×18], C23, C42 [×4], C2×C8 [×12], C22×C4 [×3], C4×C8 [×4], C2×C42, C22×C8 [×2], C2×C4×C8

Smallest permutation representation of C2×C4×C8
Regular action on 64 points
Generators in S64
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 57)(48 58)
(1 47 19 37)(2 48 20 38)(3 41 21 39)(4 42 22 40)(5 43 23 33)(6 44 24 34)(7 45 17 35)(8 46 18 36)(9 32 64 50)(10 25 57 51)(11 26 58 52)(12 27 59 53)(13 28 60 54)(14 29 61 55)(15 30 62 56)(16 31 63 49)
(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)

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

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

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

C2×C4×C8 is a maximal subgroup of
C2.C82  C42.385D4  M4(2)⋊C8  C42.46Q8  C22.7M5(2)  C42.7C8  C82⋊C2  C89M4(2)  C23.27C42  C42.455D4  C42.315D4  C42.316D4  C42.305D4  C42.42Q8  C88M4(2)  C87M4(2)  C42.43Q8  C2.C43  C424C8  (C4×C8)⋊12C4  C42.379D4  C23.17C42  Q8.C42  C42.45Q8  C4⋊C813C4  C4⋊C814C4  C8.14C42  C42.55Q8  C42.56Q8  C42.322D4  C42.58Q8  C42.59Q8  C42.60Q8  C42.324D4  C4⋊C43C8  C22⋊C44C8  C2.(C88D4)  C2.(C87D4)  C42.428D4  C42.61Q8  C87(C4⋊C4)  C85(C4⋊C4)  C42.62Q8  C42.325D4  C42.431D4  C42.432D4  C42.433D4  (C2×C4)⋊9SD16  (C2×C4)⋊6Q16  (C2×C4)⋊6D8  C42.326D4  C42.327D4  C42.436D4  C42.437D4  C4⋊M5(2)  C42.13C8  C42.6C8  C42.260C23  C42.681C23  C42.286C23  C42.290C23  C42.291C23  C42.355D4  C42.360D4  C42.364D4  C42.365D4  C42.308D4  C42.366D4  C42.367D4
C2×C4×C8 is a maximal quotient of
C82⋊C2  C424C8  C162M5(2)

64 conjugacy classes

class 1 2A···2G4A···4X8A···8AF
order12···24···48···8
size11···11···11···1

64 irreducible representations

dim11111111
type++++
imageC1C2C2C2C4C4C4C8
kernelC2×C4×C8C4×C8C2×C42C22×C8C42C2×C8C22×C4C2×C4
# reps1412416432

Matrix representation of C2×C4×C8 in GL3(𝔽17) generated by

1600
0160
0016
,
1600
0130
004
,
200
0160
009
G:=sub<GL(3,GF(17))| [16,0,0,0,16,0,0,0,16],[16,0,0,0,13,0,0,0,4],[2,0,0,0,16,0,0,0,9] >;

C2×C4×C8 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_8
% in TeX

G:=Group("C2xC4xC8");
// GroupNames label

G:=SmallGroup(64,83);
// by ID

G=gap.SmallGroup(64,83);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,48,103,117]);
// Polycyclic

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

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