C₁₇×#2312 #14525318

order	2176
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₁₇ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₁₇×#2312

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₇	C₂⁴
commutator	C₂²	C₂³×C₄×C₁₇
Frattini, cosocle	C₂³	C₂⁴×C₁₇
prime radical	C₂⁵×C₁₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ G

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

normal subgroups only

This group has 672 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) = 544

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

Conjugacy classes & character table

748 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#16927789, #16927836, #16927866, …
maximal subgroup of	C₁₇×#8736, C₁₇×#8750, C₁₇×#8759, …
maximal normal subgroup of	C₁₇×#8736, C₁₇×#8750, C₁₇×#8759, …
maximal quotient of	C₁₇×#13775, C₁₇×#13778, C₁₇×#13788, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2176, 1044);