#27422926

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #24515

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1788	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂²×#41×C₅₃	C₂²
hypercenter	#1788	D₅₃
hypocenter	C₅₃	#24515
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #1788

944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 81 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)(59,63)(62,64)(66,69,71,68)(67,73,75,76)(70,78,79,77)(72,81,80,74);(55,57)(66,68,71,69)(67,72,75,80)(70,78,79,77)(73,81,76,74);(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)(56,57)(58,60,61,65)(59,62,63,64)(68,69)(72,80)(73,76)(77,78);(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)(58,59)(60,64)(61,63)(62,65)(66,67,70,74,71,75,79,81)(68,76,77,80,69,73,78,72)

Polycyclic presentation:

2^8,53|a^2=f,b^2=e,c^2=h,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,i^a=i^52,c^b=cg,d^b=dh,i^b=i^52,f^c=fh

Conjugacy classes & character table

1034 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 208402);