#27296002

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #12071

Subgroups

	subgroup	G/H
center	C₂×C₄	D₈×D₅₃
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₄×SD₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₄×D₈
prime radical	C₂×D₈×C₅₃	C₂×C₄
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₄×SD₃₂	D₅₃
hypocenter	C₅₃	#12071
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×SD₃₂

224 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 85 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) = 6784

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

Polycyclic presentation:

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

Conjugacy classes & character table

1202 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 81478);