#34198632

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1354 ⋊ C₂
Sylow	C₇₉ ⋊ #11733

Subgroups

	subgroup	G/H
center	C₂²	#99×D₇₉
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄⋊C₄	C₂²×D₇₉
Fitting	C₇₉×#1354	C₂
socle	C₂²×C₇₉	C₂×#99
prime radical	#23766618	C₂
cosocle	C₂×C₄⋊C₄×C₇₉	C₂³
hypercenter	#1354	D₇₉
hypocenter	C₇₉	#11733
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂³×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #1354

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

normal subgroups only

This group has 632 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) = 2528

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

(1,2)(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)(88,90,92,91)(89,93,94,95);(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)(81,84)(85,86)(88,89)(90,93)(91,95)(92,94);(80,81,82,85,83,86,87,84)(89,93,94,95)

Polycyclic presentation:

2^8,79|a^2=e,c^2=g,d^2=g,e^2=h,b^a=bd,d^a=df,f^a=fh,d^b=dg,e^b=efg,i^b=i^78,i^c=i^78,e^d=eh

Conjugacy classes & character table

943 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 79931);