#34204113

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×#353×C₇₉ ⋊ C₂
Sylow	C₇₉ ⋊ #12767

Subgroups

	subgroup	G/H
center	C₂	#23763346
commutator	C₂×D₄×C₇₉	C₂²×C₄
Frattini	C₂²×D₄	C₂²×D₇₉
Fitting	C₂×#353×C₇₉	C₂
socle	C₂×C₇₉	#1790
prime radical	#23777733	C₂
cosocle	C₂²×D₄×C₇₉	C₂³
hypercenter	C₂×#353	D₇₉
hypocenter	C₇₉	#12767
LCS	G ▷ C₂×D₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×D₄ ◁ C₂×#353

878 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 878 conjugacy classes of subgroups — too many to draw. Only its 103 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) = 1264

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

(1,3)(2,7)(4,13)(5,12)(6,15)(8,16)(9,10)(11,14)(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)(2,6)(4,11)(5,12)(7,15)(8,10)(9,16)(13,14)(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)(3,10,12,7)(4,8,14,15)(5,9)(6,11)(13,16)

Polycyclic presentation:

2^8,79|a^2=f,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,e^b=egh,f^b=fg,i^b=i^78,d^c=dg,f^c=fh,i^c=i^78,e^d=eh,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, 85412);