C₁₇×#1500 #14524506

order	2176
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₇×#1500

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₇	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₁₇
Frattini, cosocle	C₂²×C₄	C₂³×C₁₇
socle	C₂²×C₁₇	C₂×C₂²⋊C₄
prime radical	C₁₇×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

238 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 41
transitive degree	μ_t(G) = 544

A minimal faithful permutation representation — 3 generators on 41 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(19,22,23,24)(26,27)(28,34,36,31)(29,32)(30,33)(35,40,41,38)(37,39);(18,19)(20,24)(21,23)(22,25)(26,28)(27,31)(29,35)(30,36)(32,38)(33,34)(37,41)(39,40);(26,29,30,37)(27,32,33,39)(28,35,36,41)(31,38,34,40)

Conjugacy classes & character table

646 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#16925364, #16925371, #16925378, …
maximal subgroup of	C₁₇×#8247, C₁₇×#8248, C₁₇×#8249, …
maximal normal subgroup of	C₁₇×#8247, C₁₇×#8248, C₁₇×#8249, …
maximal quotient of	C₁₇×#7560, C₁₇×#7562, C₁₇×#7567, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2176, 232);