#34203883

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×AΣL(1,𝔽₁₆)×C₇₉ ⋊ C₂
Sylow	C₇₉ ⋊ #12725

Subgroups

	subgroup	G/H
center	C₂	#23763316
commutator	C₂⁴×C₇₉	C₂²×C₄
Frattini	C₂²×D₄	C₂²×D₇₉
Fitting	C₂×AΣL(1,𝔽₁₆)×C₇₉	C₂
socle	C₂×C₇₉	#1782
prime radical	#23777087	C₂
cosocle	C₂²×D₄×C₇₉	C₂³
hypercenter	C₂×AΣL(1,𝔽₁₆)	D₇₉
hypocenter	C₇₉	#12725
LCS	G ▷ C₂⁴×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×D₄ ◁ C₂×AΣL(1,𝔽₁₆)

846 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 95
transitive degree	μ_t(G) = 1264

A minimal faithful permutation representation — 3 generators on 95 points:

(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)(80,82)(81,86,87,88)(83,89)(84,90)(85,93,91,94)(92,95);(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)(81,85)(86,93)(87,91)(88,94);(80,81)(82,88)(83,91,92,85)(84,87)(86,90)(89,93,95,94)

Polycyclic presentation:

2^8,79|a^2=f,c^2=g,b^a=bd,c^a=ce,d^a=dg,g^a=gh,f^b=fg,e^c=eh,i^c=i^78,f^d=fh

Conjugacy classes & character table

1048 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 85182);