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G = C2×C28order 56 = 23·7

Abelian group of type [2,28]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C28, SmallGroup(56,8)

Series: Derived Chief Lower central Upper central

C1 — C2×C28
C1C2C14C28 — C2×C28
C1 — C2×C28
C1 — C2×C28

Generators and relations for C2×C28
 G = < a,b | a2=b28=1, ab=ba >


Smallest permutation representation of C2×C28
Regular action on 56 points
Generators in S56
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 29)(25 30)(26 31)(27 32)(28 33)
(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)

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

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

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

56 conjugacy classes

class 1 2A2B2C4A4B4C4D7A···7F14A···14R28A···28X
order122244447···714···1428···28
size111111111···11···11···1

56 irreducible representations

dim11111111
type+++
imageC1C2C2C4C7C14C14C28
kernelC2×C28C28C2×C14C14C2×C4C4C22C2
# reps1214612624

Matrix representation of C2×C28 in GL2(𝔽29) generated by

280
028
,
240
02
G:=sub<GL(2,GF(29))| [28,0,0,28],[24,0,0,2] >;

C2×C28 in GAP, Magma, Sage, TeX

C_2\times C_{28}
% in TeX

G:=Group("C2xC28");
// GroupNames label

G:=SmallGroup(56,8);
// by ID

G=gap.SmallGroup(56,8);
# by ID

G:=PCGroup([4,-2,-2,-7,-2,112]);
// Polycyclic

G:=Group<a,b|a^2=b^28=1,a*b=b*a>;
// generators/relations

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