#27294409

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1354 ⋊ C₂
Sylow	C₅₃ ⋊ #11724

Subgroups

	subgroup	G/H
center	C₂²	#99×D₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄⋊C₄	C₂²×D₅₃
Fitting	C₅₃×#1354	C₂
socle	C₂²×C₅₃	C₂×#99
prime radical	C₂³×D₅₃	C₂²⋊C₄
cosocle	C₂×C₄⋊C₄×C₅₃	C₂³
hypercenter	#1354	D₅₃
hypocenter	C₅₃	#11724
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #1354

432 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 83 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

638 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 79885);