#27277321

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #8679

Subgroups

	subgroup	G/H
center	C₂²	#16810483
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2960	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂³×C₅₃	#121
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2960	D₅₃
hypocenter	C₅₃	#8679
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

350 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 101
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 101 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)(54,56,57,63)(55,60,59,68)(58,67,65,64)(61,69,66,62)(70,73,75,87)(71,77,79,85)(72,82,84,98)(74,86,81,95)(76,89,90,99)(78,91,88,100)(80,92,93,96)(83,97,94,101);(55,59)(60,68)(61,66)(62,69)(70,72,81,94,75,84,74,83)(71,76,88,93,79,90,78,80)(73,82,95,101,87,98,86,97)(77,89,100,96,85,99,91,92);(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)(56,62)(57,59)(58,66)(60,64)(61,65)(63,69)(67,68)(70,71)(72,80)(73,85,87,77)(74,88)(75,79)(76,83)(78,81)(82,96,98,92)(84,93)(86,91,95,100)(89,97,99,101)(90,94)

Polycyclic presentation:

2^8,53|a^2=f,b^2=dh,c^2=h,d^2=g,e^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,i^a=i^52,e^c=egh,f^c=fg

Conjugacy classes & character table

1502 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 62797);