#27293643

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #11548

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133132
commutator, prime radical	C₂⁴×C₅₃	C₂²×C₄
Frattini	C₂×C₂²⋊C₄	C₂²×D₅₃
Fitting	C₅₃×#2484	C₂
socle	C₂²×C₅₃	C₂×#99
cosocle	C₂×C₂²⋊C₄×C₅₃	C₂³
hypercenter	#2484	D₅₃
hypocenter	C₅₃	#11548
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ #2484

380 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 87 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)(54,57,59,56)(55,61,64,65)(58,67,68,66)(60,62,69,63)(72,78)(75,81)(79,84)(82,85);(54,56,59,57)(55,60,64,69)(58,66,68,67)(61,62,65,63)(70,72)(71,75)(73,79)(74,78)(76,82)(77,81)(80,84)(83,85);(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)(54,55,58,63,59,64,68,62)(56,65,66,69,57,61,67,60)(70,71,73,76,74,77,80,83)(72,75,79,82,78,81,84,85)

Polycyclic presentation:

2^8,53|a^2=e,b^2=h,c^2=h,e^2=g,b^a=bd,d^a=df,f^a=fh,i^a=i^52,c^b=cgh,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, 79119);