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G = C22×C14order 56 = 23·7

Abelian group of type [2,2,14]

direct product, abelian, monomial, 2-elementary

Aliases: C22×C14, SmallGroup(56,13)

Series: Derived Chief Lower central Upper central

C1 — C22×C14
C1C7C14C2×C14 — C22×C14
C1 — C22×C14
C1 — C22×C14

Generators and relations for C22×C14
 G = < a,b,c | a2=b2=c14=1, ab=ba, ac=ca, bc=cb >


Smallest permutation representation of C22×C14
Regular action on 56 points
Generators in S56
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 36)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)
(1 24)(2 25)(3 26)(4 27)(5 28)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 43)
(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)| (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56), (1,24)(2,25)(3,26)(4,27)(5,28)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,56)(42,43), (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( (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56), (1,24)(2,25)(3,26)(4,27)(5,28)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,56)(42,43), (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([(1,37),(2,38),(3,39),(4,40),(5,41),(6,42),(7,29),(8,30),(9,31),(10,32),(11,33),(12,34),(13,35),(14,36),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56)], [(1,24),(2,25),(3,26),(4,27),(5,28),(6,15),(7,16),(8,17),(9,18),(10,19),(11,20),(12,21),(13,22),(14,23),(29,44),(30,45),(31,46),(32,47),(33,48),(34,49),(35,50),(36,51),(37,52),(38,53),(39,54),(40,55),(41,56),(42,43)], [(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 2A···2G7A···7F14A···14AP
order12···27···714···14
size11···11···11···1

56 irreducible representations

dim1111
type++
imageC1C2C7C14
kernelC22×C14C2×C14C23C22
# reps17642

Matrix representation of C22×C14 in GL3(𝔽29) generated by

2800
0280
0028
,
2800
010
0028
,
1300
0250
0016
G:=sub<GL(3,GF(29))| [28,0,0,0,28,0,0,0,28],[28,0,0,0,1,0,0,0,28],[13,0,0,0,25,0,0,0,16] >;

C22×C14 in GAP, Magma, Sage, TeX

C_2^2\times C_{14}
% in TeX

G:=Group("C2^2xC14");
// GroupNames label

G:=SmallGroup(56,13);
// by ID

G=gap.SmallGroup(56,13);
# by ID

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

G:=Group<a,b,c|a^2=b^2=c^14=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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