C₁₇×#1543 #14524549

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×#1543

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₇	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₁₇
Frattini, cosocle	C₄²	C₂³×C₁₇
socle	C₂²×C₁₇	C₂×C₂²⋊C₄
prime radical	C₁₇×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

246 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 1088

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)(18,19,20,23,21,22,24,25)(26,27)(28,35)(29,33)(30,38)(31,34)(32,36)(37,40)(39,41);(19,22)(20,24)(26,28)(27,32)(29,36)(30,39)(31,37)(33,40)(34,41)(35,38);(26,29,31,30)(27,33,34,38)(28,36,37,39)(32,40,41,35)

Conjugacy classes & character table

646 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#16922073, #16922079, #16925330, …
maximal subgroup of	C₁₇×#7488, C₁₇×#7490, C₁₇×#8241, …
maximal normal subgroup of	C₁₇×#7488, C₁₇×#7490, C₁₇×#8241, …
maximal quotient of	C₁₇×#7563, C₁₇×#7567, C₁₇×#7645, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2176, 275);