C₁₇×#1742 #14524748

order	2176
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₁₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₇×#1742

Subgroups

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

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

normal subgroups only

This group has 388 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)(22,23,26,31,27,30,25,29)(24,32,33,37,34,28,35,36);(18,19,20,21)(22,24)(23,28)(25,33)(26,35)(27,34)(29,36)(30,32)(31,37);(22,25,27,26)(23,29,30,31)(24,33,34,35)(28,36,32,37)

Conjugacy classes & character table

748 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#16921453, #16939539, #16939675, …
maximal subgroup of	C₁₇×#7253, C₁₇×#10711, C₁₇×#10747, …
maximal normal subgroup of	C₁₇×#7253, C₁₇×#10711, C₁₇×#10747, …
maximal quotient of	C₁₇×#7906, C₁₇×#7907, C₁₇×#7910, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2176, 474);