#27270706

order	13568
automorphism	?
homology	Abelianization (H₁) = C₈×C₁₆ · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #7052

Subgroups

	subgroup	G/H
center, Frattini	C₈²	C₂×D₅₃
commutator	C₂×C₅₃	C₈×C₁₆
Fitting	C₈×C₁₆×C₅₃	C₂
socle, prime radical	C₂²×C₅₃	C₈²
cosocle	C₈²×C₅₃	C₂²
hypercenter	C₈×C₁₆	D₅₃
hypocenter	C₅₃	#7052
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈² ◁ C₈×C₁₆

150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 2 generators on 85 points:

(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)(54,55,56,59,57,60,62,65,58,61,63,66,64,67,68,69)(71,75)(76,81)(77,82)(83,85);(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)(70,71,72,76,73,77,78,83,74,75,79,81,80,82,84,85)

Polycyclic presentation:

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

Conjugacy classes & character table

3488 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 56182);