#27293375

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1318 ⋊ C₂
Sylow	C₅₃ ⋊ #11479

Subgroups

	subgroup	G/H
center	C₂²	#99×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄⋊C₄	C₂²×D₅₃
Fitting	C₅₃×#1318	C₂
socle	C₂²×C₅₃	C₂×#99
prime radical	C₂×#13133213	C₄
cosocle	C₂×C₄⋊C₄×C₅₃	C₂³
hypercenter	#1318	D₅₃
hypocenter	C₅₃	#11479
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #1318

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

normal subgroups only

This group has 524 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) = 77
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 77 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

722 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 78851);