C₂×#20274038 #27365255

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#20274038

Splittings

Sylow

C₅₃ ⋊ C₂×#1662

Subgroups

	subgroup	G/H
center	C₂³	#13133186
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting, prime radical	C₂×C₅₃×#562	C₂
socle	C₂³×C₅₃	#120
hypercenter	C₂×#562	D₅₃
hypocenter	C₅₃	C₂×#1662
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#562

958 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 958 conjugacy classes of subgroups — too many to draw. Only its 187 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 87 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(59,69)(62,67)(64,71)(65,68)(72,74)(73,78)(75,83)(76,85)(77,84)(79,82)(80,87)(81,86);(56,58,60,66)(57,61,63,70)(59,68,69,65)(62,71,67,64)(72,74)(73,78)(75,85)(76,83)(77,84)(79,87)(80,82)(81,86);(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)(56,57,60,63)(58,64,66,71)(59,67,69,62)(61,68,70,65)(72,73,77,81)(74,82,84,87)(75,80,76,79)(78,85,86,83);(54,55)

Conjugacy classes & character table

1028 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 150731);