C₁₇×#2372 #14525378

order	2176
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₁₇ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

C₁₇×#2372

Subgroups

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

560 conjugacy classes of subgroups; only the 280 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 280 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 — 4 generators on 37 points:

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

Conjugacy classes & character table

850 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#16931943, #16931953, #16951123, …
maximal subgroup of	C₁₇×#9439, C₁₇×#9441, C₁₇×#12965, …
maximal normal subgroup of	C₁₇×#9439, C₁₇×#9441, C₁₇×#12965, …
maximal quotient of	C₁₇×#13805, C₁₇×#13816, C₁₇×#13824, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2176, 1104);