direct product, abelian, monomial, 2-elementary
Aliases: C22×C14, SmallGroup(56,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — 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 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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