#34207398

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1390 ⋊ C₂
Sylow	C₇₉ ⋊ #13375

Subgroups

	subgroup	G/H
center	C₂²	#19098751
commutator, cosocle	C₂²×D₄×C₇₉	C₂³
Frattini	C₂²×D₄	C₂²×D₇₉
Fitting	C₇₉×#1390	C₂
socle	C₂²×C₇₉	B(4,𝔽₂)
prime radical	C₂×#14579206	C₂²
hypercenter	#1390	D₇₉
hypocenter	C₇₉	#13375
LCS	G ▷ C₂²×D₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ #1390

704 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 73 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) = 95
transitive degree	μ_t(G) = 5056

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

(9,10)(11,13)(12,14)(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)(83,84)(85,86)(87,88)(89,90)(91,92)(93,94);(1,3,4,8)(2,5,6,7)(11,15)(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)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95);(1,2,4,6)(3,7,8,5)(10,13)(14,16)

Polycyclic presentation:

2^8,79|a^2=d,b^2=d,b^a=bd,c^a=ce,f^a=fgh,c^b=cf,e^b=eg,i^b=i^78,g^c=gh,i^c=i^78,f^e=fh

Conjugacy classes & character table

1045 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 88697);