#27270995

order	13568
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #7167

Subgroups

	subgroup	G/H
center	C₂²×C₈	D₂₁₂
commutator	C₄×C₅₃	C₄×C₁₆
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂²⋊C₈
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	#7167
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

286 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 2 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)(59,63,65,64)(66,68,70,78,71,76,81,74,72,79,69,77,80,67,73,75);(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)(54,55,56,57)(58,59,61,64,62,65,60,63)(66,67,69,74,71,68,73,77,72,76,70,75,80,79,81,78)

Polycyclic presentation:

2^8,53|a^2=d,b^2=e,c^2=f,d^2=g,e^2=fg,g^2=h,b^a=bc,c^a=cf,e^a=ef,i^a=i^52

Conjugacy classes & character table

3440 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 56471);