C₂×#23761600 #34208765

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

Aliases

direct product

C₂×#23761600

Splittings

Sylow

C₇₉ ⋊ C₂×#1423

Subgroups

	subgroup	G/H
center	C₂³	D₁₂₆₄
commutator	C₁₆×C₇₉	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₇₉
Fitting	C₂²×C₃₂×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	D₁₆
cosocle	C₂×C₁₆×C₇₉	C₂³
hypercenter	C₂²×C₃₂	D₇₉
hypocenter	C₇₉	C₂×#1423
LCS	G ▷ C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂

206 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 131 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) = 117
transitive degree	μ_t(G) = 20224

A minimal faithful permutation representation — 3 generators on 117 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(86,87,93,103,89,94,106,112,100,110,114,117,113,115,99,105,91,96,108,116,101,111,98,104,90,95,107,109,97,102,88,92);(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)(82,83,84,85)(87,92)(88,93)(89,97)(90,100)(94,109)(95,112)(96,105)(98,114)(99,108)(101,113)(102,103)(104,110)(106,107)(111,117)(115,116);(80,81)

Conjugacy classes & character table

5068 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 90064);