#27299097

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #12592

Subgroups

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₅₃×#1726	C₂
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1726	D₅₃
hypocenter	C₅₃	#12592
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

358 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 97
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 3 generators on 97 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)(58,60,61,64)(59,63,62,65)(68,76)(72,82)(77,88)(78,89)(83,92)(84,93)(90,96)(94,97);(54,55,56,57)(59,62)(63,65)(66,68)(67,72)(69,77)(70,78)(71,76)(73,83)(74,84)(75,82)(79,90)(80,88)(81,89)(85,94)(86,92)(87,93)(91,96)(95,97);(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)(60,63)(61,62)(64,65)(66,67,69,73,70,74,79,85,71,75,80,86,81,87,91,95)(68,72,77,83,78,84,90,94,76,82,88,92,89,93,96,97)

Polycyclic presentation:

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

Conjugacy classes & character table

2192 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 84573);