#27270635

order	13568
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #7022

Subgroups

	subgroup	G/H
center	C₂³	#13133124
commutator	C₂×C₄×C₅₃	C₄×C₈
Frattini	C₄³	C₂×D₅₃
Fitting	C₄×C₄⋊C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	#95
cosocle	C₄³×C₅₃	C₂²
hypercenter	C₄×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#7022
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

320 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 2 generators on 77 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)(54,56,58,61,59,60,57,55)(63,66)(64,68)(71,75,77,76);(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)(62,63,64,67,65,66,68,69)(70,71,73,76,74,77,72,75)

Polycyclic presentation:

2^8,53|a^2=d,b^2=e,c^2=fg,d^2=h,e^2=fh,b^a=bc,c^a=cf,e^a=efg,i^a=i^52,d^b=dg

Conjugacy classes & character table

2144 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 56111);