#27486547

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2920 ⋊ C₂
Sylow	C₅₃ ⋊ #29384

Subgroups

	subgroup	G/H
center	C₄	#16811187
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2920	C₂
socle	C₂×C₅₃	#2598
prime radical	#20275782	C₂
hypercenter	#2920	D₅₃
hypocenter	C₅₃	#29384
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2920

708 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

1502 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 272023);