C₁₇×#1740 #14524746

order	2176
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₁₇ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₁₇×#1740

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₇	C₂²⋊C₄
commutator	C₂²	C₂×C₄²×C₁₇
Frattini, cosocle	C₂⁴	C₂³×C₁₇
socle	C₂²×C₁₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₇	C₂³
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₇ ◁ C₂³×C₄×C₁₇ ◁ G

420 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 216 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) = 544

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)(22,23)(24,30,32,27)(25,28)(26,29)(31,36,37,34)(33,35);(18,19,20,21)(22,24)(23,27)(25,31)(26,32)(28,34)(29,30)(33,37)(35,36);(22,25,26,33)(23,28,29,35)(24,31,32,37)(27,34,30,36)

Conjugacy classes & character table

748 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#16921450, #16939535, #16939787, …
maximal subgroup of	C₁₇×#7252, C₁₇×#10711, C₁₇×#10772, …
maximal normal subgroup of	C₁₇×#7252, C₁₇×#10711, C₁₇×#10772, …
maximal quotient of	C₁₇×#7906, C₁₇×#7908, C₁₇×#7910, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2176, 472);