#34203461

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #12632

Subgroups

	subgroup	G/H
center	C₂²	#99×D₇₉
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂²×Q₈	C₂²×D₇₉
Fitting	C₂×#355×C₇₉	C₂
socle	C₂²×C₇₉	C₂×#99
prime radical	C₂²×D₄×C₇₉	C₂×C₄
cosocle	C₂²×Q₈×C₇₉	C₂³
hypercenter	C₂×#355	D₇₉
hypocenter	C₇₉	#12632
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ C₂×#355

512 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 125 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) = 99
transitive degree	μ_t(G) = 5056

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

Polycyclic presentation:

2^8,79|a^2=e,b^2=f,c^2=eh,d^2=h,f^2=h,b^a=bd,c^a=ch,d^a=dh,f^a=fgh,i^a=i^78,d^b=dg,g^b=gh,i^c=i^78,f^d=fh

Conjugacy classes & character table

1060 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 84760);