#34495196

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×#120×C₇₉ ⋊ C₂
Sylow	C₇₉ ⋊ #36714

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₇₉
commutator, socle, cosocle	C₂³×C₇₉	C₂⁵
Fitting	C₂²×#120×C₇₉	C₂
hypercenter	C₂²×#120	D₇₉
hypocenter	C₇₉	#36714
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×#120

4170 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 99
transitive degree	μ_t(G) = 2528

A minimal faithful permutation representation — 5 generators on 99 points:

(6,10)(9,11)(14,18)(17,20)(21,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23);(6,10)(7,12)(15,19)(17,20);(2,4)(15,19)(17,20);(5,7)(6,9)(8,12)(10,11)(13,15)(14,17)(16,19)(18,20);(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,17)(16,18)(19,20)(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)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97)(98,99)

Polycyclic presentation:

2^8,79|a^2=g,b^a=bf,c^a=cg,d^a=df,e^a=efh,i^a=i^78,c^b=ch,d^b=dfh

Conjugacy classes & character table

2236 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 376495);