#34197354

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1317 ⋊ C₂
Sylow	C₇₉ ⋊ #11425

Subgroups

	subgroup	G/H
center	C₂²	#99×D₇₉
commutator	C₂³×C₇₉	C₂²×C₈
Frattini	C₂×C₂²⋊C₄	C₂²×D₇₉
Fitting	C₇₉×#1317	C₂
socle	C₂²×C₇₉	C₂×#99
prime radical	C₂⁴×D₇₉	C₂×C₄
cosocle	C₂×C₂²⋊C₄×C₇₉	C₂³
hypercenter	#1317	D₇₉
hypocenter	C₇₉	#11425
LCS	G ▷ C₂³×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₂²⋊C₄ ◁ #1317

678 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 93 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) = 103
transitive degree	μ_t(G) = 2528

A minimal faithful permutation representation — 3 generators on 103 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)(88,90,92,97)(89,94,95,101)(91,96,99,103)(93,100,98,102);(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)(89,93)(94,100)(95,98)(101,102);(80,81,82,84,83,85,86,87)(88,89)(90,94)(91,98,99,93)(92,95)(96,102,103,100)(97,101)

Polycyclic presentation:

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

Conjugacy classes & character table

1066 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 78653);