#27490247

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3459 ⋊ C₂
Sylow	C₅₃ ⋊ #29686

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133186
commutator, cosocle	C₂⁴×C₅₃	C₂⁴
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₅₃×#3459	C₂
socle	C₂²×C₅₃	C₂×#120
hypercenter	#3459	D₅₃
hypocenter	C₅₃	#29686
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #3459

1312 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1312 conjugacy classes of subgroups — too many to draw. Only its 187 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) = 1696

A minimal faithful permutation representation — 4 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)(54,56,57,60)(55,59,58,61)(65,73)(68,76)(71,77)(74,75);(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)(56,60)(59,61)(62,65)(63,68)(64,71)(66,73)(67,74)(69,76)(70,75)(72,77);(55,58)(59,61)(62,64)(63,67)(65,71)(66,72)(68,75)(69,70)(73,77)(74,76);(54,55)(56,59)(57,58)(60,61)(62,63)(64,70)(65,68)(66,69)(67,72)(71,74)(73,76)(75,77)

Polycyclic presentation:

2^8,53|d^2=eh,b^a=be,c^a=cf,g^a=gh,c^b=cg,f^b=fh,d^c=de,i^c=i^52

Conjugacy classes & character table

1181 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 275723);