C₂×#23761422 #34198541

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#23761422

Splittings

Sylow

C₇₉ ⋊ C₂×#1351

Subgroups

	subgroup	G/H
center	C₂³	#14579075
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₂²⋊C₄	C₂²×D₇₉
Fitting	C₂×#323×C₇₉	C₂
socle	C₂³×C₇₉	#99
prime radical	C₂⁵×C₇₉	D₄
cosocle	C₂×C₂²⋊C₄×C₇₉	C₂³
hypercenter	C₂×#323	D₇₉
hypocenter	C₇₉	C₂×#1351
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂³×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ C₂×#323

776 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 123 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) = 97
transitive degree	μ_t(G) = 5056

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

Conjugacy classes & character table

1762 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 79840);