C₂×#20272876 #27304376

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂×#20272876

Splittings

Sylow

C₅₃ ⋊ C₂×#1407

Subgroups

	subgroup	G/H
center	C₂²	#16810415
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	#124	C₂²×D₅₃
Fitting	C₂×C₅₃×#355	C₂
socle	C₂²×C₅₃	#353
prime radical	C₂²×D₄×C₅₃	D₄
cosocle	#124×C₅₃	C₂³
hypercenter	C₂×#355	D₅₃
hypocenter	C₅₃	C₂×#1407
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#124 ◁ C₂×#355

416 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 75 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) = 3392

A minimal faithful permutation representation — 3 generators on 87 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 89852);