C₁₇×#1738 #14524744

order	2176
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₁₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₇×#1738

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₇	C₂×D₄
commutator	C₄	C₂×C₄²×C₁₇
Frattini, cosocle	C₂²×C₄	C₂³×C₁₇
socle	C₂²×C₁₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₇	Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₇ ◁ C₂³×C₄×C₁₇ ◁ G

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

normal subgroups only

This group has 372 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)(18,19)(20,21)(22,23,25,30,27,31,26,29)(24,32,34,28,33,37,35,36);(19,21)(22,24,26,35,27,33,25,34)(23,28,30,37,31,36,29,32);(24,33)(28,36)(32,37)(34,35)

Conjugacy classes & character table

748 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#16939660, #16939705, #16939768, …
maximal subgroup of	C₁₇×#10744, C₁₇×#10753, C₁₇×#10767, …
maximal normal subgroup of	C₁₇×#10744, C₁₇×#10753, C₁₇×#10767, …
maximal quotient of	C₁₇×#7873, C₁₇×#7876, C₁₇×#7877, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2176, 470);