#34183757

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #9098

Subgroups

	subgroup	G/H
center	C₂×C₄	#14579135
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#2965	C₂
socle	C₂²×C₇₉	#386
prime radical	C₂×#41×C₇₉	C₂×C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#2965	D₇₉
hypocenter	C₇₉	#9098
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2965

488 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 103
transitive degree	μ_t(G) = 10112

A minimal faithful permutation representation — 3 generators on 103 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(81,82)(86,87)(88,90)(89,94)(91,99)(92,101)(93,95)(96,102)(97,103)(98,100);(80,82,84,87,83,81,85,86)(88,90)(89,93,96,97)(91,99)(92,101)(94,95,102,103)(98,100);(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)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81,84,86,83,82,85,87)(88,89)(90,97,99,93)(91,96)(92,95)(94,98,102,101)(100,103)

Polycyclic presentation:

2^8,79|a^2=f,b^2=dfg,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,i^a=i^78,c^b=ch,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=eg,f^c=fg

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, 65056);