#34177116

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

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #7761

Subgroups

	subgroup	G/H
center	C₂³	#14579129
commutator, cosocle	C₂×C₄²×C₇₉	C₂³
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#2058	C₂
socle, prime radical	C₂³×C₇₉	#120
hypercenter	#2058	D₇₉
hypocenter	C₇₉	#7761
LCS	G ▷ C₂×C₄²×C₇₉ ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2058

436 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 20224

A minimal faithful permutation representation — 3 generators on 111 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)(80,82,83,86)(81,84,85,87)(88,89,91,93)(90,95,94,92);(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)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,84)(83,85)(86,87)(96,98,100,106)(97,101,103,110)(99,108,109,105)(102,111,107,104);(84,87)(89,92)(90,94)(93,95)(96,97,100,103)(98,104,106,111)(99,107,109,102)(101,108,110,105)

Polycyclic presentation:

2^8,79|a^2=f,b^2=f,c^2=gh,d^2=f,e^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,d^b=df,e^b=eg,i^b=i^78,d^c=dg,e^c=eh

Conjugacy classes & character table

1285 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 58415);