C₂×#23761591 #34208737

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

Aliases

direct product

C₂×#23761591

Splittings

Fitting	C₂×M₆₄×C₇₉ ⋊ C₂
Sylow	C₇₉ ⋊ C₂×#1419

Subgroups

	subgroup	G/H
center	C₂²	#19098365
commutator	C₁₆×C₇₉	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₇₉
Fitting	C₂×M₆₄×C₇₉	C₂
socle	C₂²×C₇₉	#357
prime radical	C₂²×D₁₂₆₄	C₂
cosocle	C₂×C₁₆×C₇₉	C₂³
hypercenter	C₂×M₆₄	D₇₉
hypocenter	C₇₉	C₂×#1419
LCS	G ▷ C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×M₆₄

326 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 107 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) = 113
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 3 generators on 113 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)(83,89)(84,95)(85,87)(86,102)(90,108)(91,111)(92,110)(93,94)(96,103)(97,107)(98,100)(99,113)(101,106)(104,105)(109,112);(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)(82,83,85,91,88,93,87,104)(84,94,97,111,100,89,99,105)(86,101,96,108,103,109,102,92)(90,107,106,95,110,113,112,98);(80,81)

Conjugacy classes & character table

3172 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 90036);