#27270715

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #7056

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133132
commutator	C₂×C₄×C₅₃	C₂×C₁₆
Frattini	C₂×C₄⋊C₈	C₂×D₅₃
Fitting	C₅₃×#1954	C₂
socle	C₂²×C₅₃	#323
prime radical	C₂²×D₄×C₅₃	C₈
cosocle	C₂×C₄⋊C₈×C₅₃	C₂²
hypercenter	#1954	D₅₃
hypocenter	C₅₃	#7056
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #1954

248 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 59 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 — 2 generators on 77 points:

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

Polycyclic presentation:

2^8,53|a^2=d,c^2=g,d^2=f,f^2=h,b^a=bc,c^a=ce,e^a=eg,i^a=i^52,c^b=cg,d^b=deg,d^c=dg

Conjugacy classes & character table

1508 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 56191);