#34177681

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #7858

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₇₉
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#2283	C₂
socle	C₂²×C₇₉	C₂×#95
prime radical	C₂⁴×C₇₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#2283	D₇₉
hypocenter	C₇₉	#7858
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ #2283

592 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 5056

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)(80,83,85,84)(81,87,89,88)(82,90,91,92)(86,93,94,95);(80,82)(81,86)(83,92)(84,90)(85,91)(87,95)(88,93)(89,94)(96,98)(97,101,103,108)(99,105)(100,106)(102,104,109,110)(107,111);(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)(80,81,84,88,85,89,83,87)(82,86,90,93,91,94,92,95)(96,97)(98,104,106,110)(99,102)(100,103)(101,105,108,111)(107,109)

Polycyclic presentation:

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

Conjugacy classes & character table

2248 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 58980);