#34195066

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₂×#390×C₇₉ ⋊ C₂
Sylow	C₇₉ ⋊ #11022

Subgroups

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₇₉
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₉
Fitting	C₂×#390×C₇₉	C₂
socle	C₂⁴×C₇₉	C₂×D₄
prime radical	C₂³×#11929609	C₂
cosocle	C₂³×C₄×C₇₉	C₂³
hypercenter	C₂×#390	D₇₉
hypocenter	C₇₉	#11022
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#390

1822 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 99
transitive degree	μ_t(G) = 10112

A minimal faithful permutation representation — 3 generators on 99 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)(89,91)(93,97)(94,95)(98,99);(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)(85,87)(92,93)(94,98)(95,97)(96,99);(80,81,82,83)(84,85)(86,87)(88,89)(90,91)(93,97)(94,95)(98,99)

Polycyclic presentation:

2^8,79|a^2=g,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=dh,f^b=fh,i^b=i^78,d^c=dh,f^c=fh

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, 76365);