C₂×#20274016 #27365276

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

Aliases

direct product

C₂×#20274016

Splittings

Sylow

C₅₃ ⋊ C₂×#1658

Subgroups

	subgroup	G/H
center	C₂³	#13133185
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#490	C₂
socle	C₂³×C₅₃	#120
prime radical	C₂×#127×C₅₃	C₂²
hypercenter	C₂×#490	D₅₃
hypocenter	C₅₃	C₂×#1658
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#490

882 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 882 conjugacy classes of subgroups — too many to draw. Only its 171 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:

(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)(56,58,60,65)(57,62,63,71)(59,68,66,64)(61,69,67,70)(72,74)(73,77)(75,84)(76,81)(78,87)(79,86)(80,82)(83,85);(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)(56,57,60,63)(58,62,65,71)(59,67,66,61)(64,69,68,70)(72,73,76,79)(74,77,81,86)(75,83,82,78)(80,87,84,85);(57,61)(59,66)(62,70)(63,67)(64,68)(69,71)(74,80)(75,82)(77,85)(78,83)(81,84)(86,87);(54,55)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 150752);