#27293783

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #11579

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133132
commutator	C₂⁴×C₅₃	C₂²×C₄
Frattini	C₂×C₂²⋊C₄	C₂²×D₅₃
Fitting	C₅₃×#2487	C₂
socle	C₂²×C₅₃	C₂×#99
prime radical	C₂²×D₄×C₅₃	C₂×C₄
cosocle	C₂×C₂²⋊C₄×C₅₃	C₂³
hypercenter	#2487	D₅₃
hypocenter	C₅₃	#11579
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ #2487

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

normal subgroups only

This group has 424 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) = 85
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 85 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)(70,72,74,78)(71,76,77,84)(73,79,81,85)(75,82,80,83);(54,56)(55,59,62,69)(57,65)(58,66)(60,68,61,63)(64,67)(71,75,77,80)(72,78)(73,81)(76,83,84,82);(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)(54,55,57,61,58,62,67,60)(56,63,65,59,66,68,64,69)(70,71)(72,76)(73,80,81,75)(74,77)(78,84)(79,83,85,82)

Polycyclic presentation:

2^8,53|a^2=e,b^2=f,c^2=gh,e^2=g,b^a=bd,d^a=df,f^a=fh,i^a=i^52,c^b=cgh,d^b=dh,e^b=ef,e^d=eh

Conjugacy classes & character table

1025 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 79259);