C₁₇×#1890 #14524896

order	2176
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₇×#1890

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₇	#120
commutator	C₂³	C₂²×C₄×C₁₇
Frattini, cosocle	C₂⁴	C₂³×C₁₇
prime radical	C₂²×D₄×C₁₇	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₇ ◁ C₂⁴×C₁₇ ◁ G

386 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 112 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)(19,21)(22,23)(24,27)(25,32,33,28)(26,29)(30,34,36,37)(31,35);(18,19,20,21)(22,24,26,31)(23,27,29,35)(25,30,33,36)(28,34,32,37);(22,25)(23,28)(24,30)(26,33)(27,34)(29,32)(31,36)(35,37)

Conjugacy classes & character table

544 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#16950088, #16950128, #16950138, …
maximal subgroup of	C₁₇×#12754, C₁₇×#12760, C₁₇×#12761, …
maximal normal subgroup of	C₁₇×#12754, C₁₇×#12760, C₁₇×#12761, …
maximal quotient of	C₁₇×#8966, C₁₇×#8968, C₁₇×#9011, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2176, 622);