#34182987

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

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1276 ⋊ C₂
Sylow	C₇₉ ⋊ #8971

Subgroups

	subgroup	G/H
center	C₂×C₄	#14579131
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1276	C₂
socle	C₂²×C₇₉	#386
prime radical	C₂×#11929624	D₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1276	D₇₉
hypocenter	C₇₉	#8971
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1276

478 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 99 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) = 5056

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)(89,94)(91,98)(92,100)(93,101)(95,103)(96,102)(97,99);(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)(80,81,82,86,83,85,87,84)(89,93)(92,99)(94,101)(97,100);(81,84,85,86)(88,89)(90,96,98,103)(91,93)(92,95)(94,97,101,100)(99,102)

Polycyclic presentation:

2^8,79|a^2=f,b^2=dfh,d^2=g,e^2=h,b^a=bd,c^a=ce,d^b=dg,e^b=eh,f^b=fg,i^b=i^78,d^c=dh,e^c=eh,f^c=fh,i^c=i^78

Conjugacy classes & character table

2236 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 64286);