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## G = C2×C4×C8order 64 = 26

### Abelian group of type [2,4,8]

Aliases: C2×C4×C8, SmallGroup(64,83)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4×C8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C2×C4×C8
 Lower central C1 — C2×C4×C8
 Upper central C1 — C2×C4×C8
 Jennings C1 — C2 — C2 — C2×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, C4, C22, C22, C8, C2×C4, C2×C4, C23, C42, C2×C8, C22×C4, C22×C4, C4×C8, C2×C42, C22×C8, C2×C4×C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C42, C2×C8, C22×C4, C4×C8, C2×C42, C22×C8, C2×C4×C8

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

64 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4X 8A ··· 8AF order 1 2 ··· 2 4 ··· 4 8 ··· 8 size 1 1 ··· 1 1 ··· 1 1 ··· 1

64 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + + + image C1 C2 C2 C2 C4 C4 C4 C8 kernel C2×C4×C8 C4×C8 C2×C42 C22×C8 C42 C2×C8 C22×C4 C2×C4 # reps 1 4 1 2 4 16 4 32

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

 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
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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