#27440911

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2028 ⋊ C₂
Sylow	C₅₃ ⋊ #25927

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2028	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	C₂×#11719945	C₂³
hypercenter	#2028	D₅₃
hypocenter	C₅₃	#25927
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2028

950 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 93 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)(54,57,58,56)(55,61,60,59)(63,71)(65,79)(69,73)(72,81)(74,92)(78,93)(80,83)(85,86);(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)(55,60)(59,61)(62,65,68,83,66,79,67,80)(63,70,81,89,71,88,72,77)(64,74,82,78,75,92,76,93)(69,84,86,91,73,90,85,87);(54,56,58,57)(55,59,60,61)(62,64,66,75)(63,69,71,73)(65,78,79,93)(67,76,68,82)(70,87,88,91)(72,85,81,86)(74,83,92,80)(77,90,89,84);(54,55,58,60)(56,61,57,59)(62,63,66,71)(64,73,75,69)(65,77,79,89)(67,81,68,72)(70,83,88,80)(74,91,92,87)(76,85,82,86)(78,84,93,90)

Polycyclic presentation:

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

Conjugacy classes & character table

716 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 226387);