#27271088

order	13568
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #7203

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133131
commutator	C₂³×C₅₃	C₄×C₈
Frattini	C₂×C₄×D₄	C₂×D₅₃
Fitting	C₄×#100×C₅₃	C₂
socle	C₂²×C₅₃	#342
prime radical	C₂²×D₄×C₅₃	C₂×C₄
cosocle	C₂×C₄×D₄×C₅₃	C₂²
hypercenter	C₄×#100	D₅₃
hypocenter	C₅₃	#7203
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ C₄×#100

364 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 95 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) = 69
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 2 generators on 69 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)(54,56,58,61,59,60,57,55)(63,66)(67,68);(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)(54,55,57,60,59,61,58,56)(62,63)(64,68,69,66)(65,67)

Polycyclic presentation:

2^8,53|a^2=d,b^2=e,d^2=g,e^2=g,b^a=bc,c^a=cf,f^a=fh,i^a=i^52,d^b=df,i^b=i^52,d^c=dh

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 56564);