#27294426

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #11727

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133132
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₂²⋊C₄	C₂²×D₅₃
Fitting	C₅₃×#2395	C₂
socle	C₂²×C₅₃	C₂×#99
prime radical	C₂²×D₄×C₅₃	D₄
cosocle	C₂×C₂²⋊C₄×C₅₃	C₂³
hypercenter	#2395	D₅₃
hypocenter	C₅₃	#11727
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ #2395

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

normal subgroups only

This group has 456 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) = 77
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 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)(54,56,58,57)(55,59,60,61);(54,55)(56,59)(57,61)(58,60)(62,64,66,72)(63,67,69,75)(65,71,74,77)(68,70,73,76);(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)(55,59,60,61)(62,63)(64,70)(65,73,74,68)(66,69)(67,71,75,77)(72,76)

Polycyclic presentation:

2^8,53|a^2=e,b^2=h,c^2=g,d^2=g,b^a=bd,d^a=df,f^a=fh,i^a=i^52,d^b=dg,e^b=efg,e^d=eh

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 79902);