#27293523

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1327 ⋊ C₂
Sylow	C₅₃ ⋊ #11521

Subgroups

	subgroup	G/H
center	C₄	#16810595
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×M₁₆	C₂²×D₅₃
Fitting	C₅₃×#1327	C₂
socle	C₂×C₅₃	#1464
prime radical	C₂³×D₅₃	M₁₆
cosocle	C₂×M₁₆×C₅₃	C₂³
hypercenter	#1327	D₅₃
hypocenter	C₅₃	#11521
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #1327

330 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 69 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) = 69
transitive degree	μ_t(G) = 848

A minimal faithful permutation representation — 3 generators on 69 points:

(1,2)(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)(55,60,59,62)(56,63)(61,69,66,68)(64,67);(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)(60,62);(54,55,56,61,57,62,64,69,58,59,63,66,65,60,67,68)

Polycyclic presentation:

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

Conjugacy classes & character table

716 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 78999);