C₂×#23761552 #34208517

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

Aliases

direct product

C₂×#23761552

Splittings

Fitting	C₂×#329×C₇₉ ⋊ C₂
Sylow	C₇₉ ⋊ C₂×#1404

Subgroups

	subgroup	G/H
center	C₂²	#19098351
commutator	C₂×Q₈×C₇₉	C₂²×C₄
Frattini	#124	C₂²×D₇₉
Fitting	C₂×#329×C₇₉	C₂
socle	C₂²×C₇₉	AΣL(1,𝔽₁₆)
prime radical	C₂×#19098735	C₂
cosocle	#124×C₇₉	C₂³
hypercenter	C₂×#329	D₇₉
hypocenter	C₇₉	C₂×#1404
LCS	G ▷ C₂×Q₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#124 ◁ C₂×#329

510 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 75 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) = 2528

A minimal faithful permutation representation — 3 generators on 97 points:

(3,4)(5,12)(6,15)(7,11)(8,14)(9,13)(10,17)(16,18)(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)(95,96);(3,4,5,9)(6,15,16,12)(7,11,14,18)(8,17,10,13)(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)(96,97);(1,2)

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, 89816);