C₁₇×#1815 #14524821

order	2176
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₇×#1815

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₇	#121
commutator	C₂×C₄	C₂²×C₄×C₁₇
Frattini, cosocle	C₂²×C₄	C₂³×C₁₇
prime radical	C₂×#41×C₁₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₇ ◁ C₂²×C₄×C₁₇ ◁ G

264 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 37
transitive degree	μ_t(G) = 1088

A minimal faithful permutation representation — 3 generators on 37 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,21)(22,23,27,30,28,25,26,36)(24,31,34,32,35,37,33,29);(22,24,26,33,28,35,27,34)(23,29,30,31,25,32,36,37);(19,21)(22,25)(23,28)(24,32)(26,36)(27,30)(29,35)(31,34)(33,37)

Conjugacy classes & character table

544 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#16950074, #16950487, #16950900, …
maximal subgroup of	C₁₇×#12752, C₁₇×#12833, C₁₇×#12933, …
maximal normal subgroup of	C₁₇×#12752, C₁₇×#12833, C₁₇×#12933, …
maximal quotient of	C₁₇×#8604, C₁₇×#8607, C₁₇×#8626, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2176, 547);