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G = C56⋊C2order 112 = 24·7

2nd semidirect product of C56 and C2 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C82D7, C562C2, C71SD16, C4.8D14, C2.3D28, C14.1D4, D28.1C2, Dic141C2, C28.8C22, SmallGroup(112,5)

Series: Derived Chief Lower central Upper central

C1C28 — C56⋊C2
C1C7C14C28D28 — C56⋊C2
C7C14C28 — C56⋊C2
C1C2C4C8

Generators and relations for C56⋊C2
 G = < a,b | a56=b2=1, bab=a27 >

28C2
14C22
14C4
4D7
7Q8
7D4
2Dic7
2D14
7SD16

Smallest permutation representation of C56⋊C2
On 56 points
Generators in S56
(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)
(2 28)(3 55)(4 26)(5 53)(6 24)(7 51)(8 22)(9 49)(10 20)(11 47)(12 18)(13 45)(14 16)(15 43)(17 41)(19 39)(21 37)(23 35)(25 33)(27 31)(30 56)(32 54)(34 52)(36 50)(38 48)(40 46)(42 44)

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

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

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

C56⋊C2 is a maximal subgroup of
D567C2  C8⋊D14  C8.D14  D8⋊D7  D7×SD16  SD163D7  Q16⋊D7  C56⋊C6  C6.D28  C21⋊SD16  C8⋊D21
C56⋊C2 is a maximal quotient of
C28.44D4  C8⋊Dic7  C2.D56  C6.D28  C21⋊SD16  C8⋊D21

31 conjugacy classes

class 1 2A2B4A4B7A7B7C8A8B14A14B14C28A···28F56A···56L
order122447778814141428···2856···56
size1128228222222222···22···2

31 irreducible representations

dim1111222222
type++++++++
imageC1C2C2C2D4D7SD16D14D28C56⋊C2
kernelC56⋊C2C56Dic14D28C14C8C7C4C2C1
# reps11111323612

Matrix representation of C56⋊C2 in GL4(𝔽113) generated by

1310000
131300
001112
0011103
,
1000
011200
001031
001410
G:=sub<GL(4,GF(113))| [13,13,0,0,100,13,0,0,0,0,1,11,0,0,112,103],[1,0,0,0,0,112,0,0,0,0,103,14,0,0,1,10] >;

C56⋊C2 in GAP, Magma, Sage, TeX

C_{56}\rtimes C_2
% in TeX

G:=Group("C56:C2");
// GroupNames label

G:=SmallGroup(112,5);
// by ID

G=gap.SmallGroup(112,5);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-7,61,26,182,42,2404]);
// Polycyclic

G:=Group<a,b|a^56=b^2=1,b*a*b=a^27>;
// generators/relations

Export

Subgroup lattice of C56⋊C2 in TeX

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