C₂×#23761533 #34208441

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#23761533

Splittings

Sylow

C₇₉ ⋊ C₂×#1397

Subgroups

	subgroup	G/H
center	C₂²×C₁₆	C₂×D₇₉
commutator	C₂×C₇₉	C₂×C₄×C₁₆
Frattini	C₂×C₁₆	C₂²×D₇₉
Fitting	C₂²×C₃₂×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₁₆
cosocle	C₂×C₁₆×C₇₉	C₂³
hypercenter	C₂²×C₃₂	D₇₉
hypocenter	C₇₉	C₂×#1397
LCS	G ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂

254 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 179 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:

(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)(93,102)(94,103)(95,104)(105,112)(106,113)(107,114)(115,117);(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)(86,87,88,93,89,94,96,105,90,95,97,106,99,107,108,115,91,92,98,102,100,103,109,112,101,104,110,113,111,114,116,117);(80,81)

Conjugacy classes & character table

5152 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 89740);