#27538113

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3018 ⋊ C₂
Sylow	C₅₃ ⋊ #33546

Subgroups

	subgroup	G/H
center	C₄	#16810809
commutator, cosocle	C₂×D₄×C₅₃	C₂⁴
Frattini	C₂×D₄	C₂³×D₅₃
Fitting	C₅₃×#3018	C₂
socle	C₂×C₅₃	C₂×B(4,𝔽₂)
hypercenter	#3018	D₅₃
hypocenter	C₅₃	#33546
LCS	G ▷ C₂×D₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×#41 ◁ #3018

1236 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 1236 conjugacy classes of subgroups — too many to draw. Only its 151 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) = 848

A minimal faithful permutation representation — 4 generators on 69 points:

(1,3,5,10)(2,7,8,14)(4,9,11,15)(6,12,13,16)(17,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19);(4,11)(9,15)(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)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69);(2,6)(7,12)(8,13)(14,16)(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)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69);(1,2)(3,7)(4,6)(5,8)(9,12)(10,14)(11,13)(15,16)

Polycyclic presentation:

2^8,53|d^2=h,b^a=be,c^a=cf,g^a=gh,f^b=fgh,i^b=i^52,e^c=eg,i^c=i^52,f^e=fh

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323589);