#27299603

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1785 ⋊ C₂
Sylow	C₅₃ ⋊ #12701

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133131
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×Q₈	C₂²×D₅₃
Fitting	C₅₃×#1785	C₂
socle	C₂²×C₅₃	C₂×#99
prime radical	C₂×#16811169	C₂
cosocle	C₂²×Q₈×C₅₃	C₂³
hypercenter	#1785	D₅₃
hypocenter	C₅₃	#12701
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ #1785

1070 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1070 conjugacy classes of subgroups — too many to draw. Only its 115 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) = 73
transitive degree	μ_t(G) = 1696

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

Polycyclic presentation:

2^8,53|a^2=f,c^2=h,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dh,f^b=fgh,i^b=i^52,f^d=fh

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 85079);