Copied to
clipboard

G = C22xC14order 56 = 23·7

Abelian group of type [2,2,14]

direct product, abelian, monomial, 2-elementary

Aliases: C22xC14, SmallGroup(56,13)

Series: Derived Chief Lower central Upper central

C1 — C22xC14
C1C7C14C2xC14 — C22xC14
C1 — C22xC14
C1 — C22xC14

Generators and relations for C22xC14
 G = < a,b,c | a2=b2=c14=1, ab=ba, ac=ca, bc=cb >

Subgroups: 32, all normal (4 characteristic)
Quotients: C1, C2, C22, C7, C23, C14, C2xC14, C22xC14

Smallest permutation representation of C22xC14
Regular action on 56 points
Generators in S56
(1 41)(2 42)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 55)(16 56)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)
(1 28)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(14 27)(29 56)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(37 50)(38 51)(39 52)(40 53)(41 54)(42 55)
(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)| (1,41)(2,42)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,55)(16,56)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54), (1,28)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(29,56)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52)(40,53)(41,54)(42,55), (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( (1,41)(2,42)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,55)(16,56)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54), (1,28)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(29,56)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52)(40,53)(41,54)(42,55), (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([[(1,41),(2,42),(3,29),(4,30),(5,31),(6,32),(7,33),(8,34),(9,35),(10,36),(11,37),(12,38),(13,39),(14,40),(15,55),(16,56),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,49),(24,50),(25,51),(26,52),(27,53),(28,54)], [(1,28),(2,15),(3,16),(4,17),(5,18),(6,19),(7,20),(8,21),(9,22),(10,23),(11,24),(12,25),(13,26),(14,27),(29,56),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,49),(37,50),(38,51),(39,52),(40,53),(41,54),(42,55)], [(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)]])

C22xC14 is a maximal subgroup of   C23.D7  C7.F8

56 conjugacy classes

class 1 2A···2G7A···7F14A···14AP
order12···27···714···14
size11···11···11···1

56 irreducible representations

dim1111
type++
imageC1C2C7C14
kernelC22xC14C2xC14C23C22
# reps17642

Matrix representation of C22xC14 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] >;

C22xC14 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

Export

Subgroup lattice of C22xC14 in TeX

׿
x
:
Z
F
o
wr
Q
<