#34176012

order	20224
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #7486

Subgroups

	subgroup	G/H
center	C₄	#19098351
commutator	C₂×D₄×C₇₉	C₄²
Frattini	#532	C₂×D₇₉
Fitting	C₇₉×#1525	C₂
socle	C₂×C₇₉	#1390
prime radical, cosocle	#532×C₇₉	C₂²
hypercenter	#1525	D₇₉
hypocenter	C₇₉	#7486
LCS	G ▷ C₂×D₄×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ #1525

286 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 59 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 — 2 generators on 95 points:

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

Polycyclic presentation:

2^8,79|a^2=d,b^2=e,e^2=h,f^2=h,b^a=bc,c^a=cf,f^a=fg,g^a=gh,i^a=i^78,d^b=dfh,f^b=fh,g^b=gh,i^b=i^78,d^c=dgh,f^c=fh,f^d=fh

Conjugacy classes & character table

1516 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 57311);