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## G = C22×C14order 56 = 23·7

### Abelian group of type [2,2,14]

Aliases: C22×C14, SmallGroup(56,13)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C14
 Chief series C1 — C7 — C14 — C2×C14 — C22×C14
 Lower central C1 — C22×C14
 Upper central 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 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)]])

C22×C14 is a maximal subgroup of   C23.D7  C7.F8

56 conjugacy classes

 class 1 2A ··· 2G 7A ··· 7F 14A ··· 14AP order 1 2 ··· 2 7 ··· 7 14 ··· 14 size 1 1 ··· 1 1 ··· 1 1 ··· 1

56 irreducible representations

 dim 1 1 1 1 type + + image C1 C2 C7 C14 kernel C22×C14 C2×C14 C23 C22 # reps 1 7 6 42

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

 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
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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