#27275249

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #8266

Subgroups

	subgroup	G/H
center	C₄²	D₄×D₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1276	C₂
socle	C₂²×C₅₃	C₄×C₄⋊C₄
prime radical	C₂³×C₅₃	#117
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1276	D₅₃
hypocenter	C₅₃	#8266
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1276

414 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 175 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)(62,64,65,70,66,71,73,77)(63,68,72,76,69,75,67,74);(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)(54,55,56,58,57,59,60,61)(62,63)(64,68)(65,72)(66,69)(67,73)(70,76)(71,75)(74,77);(63,67,69,72)(68,74,75,76)

Polycyclic presentation:

2^8,53|a^2=e,b^2=f,c^2=de,d^2=g,f^2=h,b^a=bd,d^b=dg,e^b=eg,i^c=i^52

Conjugacy classes & character table

1568 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 60725);