C₁₇×#1564 #14524570

order	2176
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₇×#1564

Subgroups

	subgroup	G/H
center	C₄²×C₁₇	D₄
commutator	C₄	C₂²×C₈×C₁₇
Frattini, cosocle	C₄²	C₂³×C₁₇
socle	C₂²×C₁₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₇	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

180 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 128 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)(18,19,20,22,21,23,24,25)(26,27)(28,35)(29,33)(30,39)(31,34)(32,37)(36,40)(38,41);(26,28,30,37,31,38,29,36)(27,32,33,35,34,40,39,41);(26,29,31,30)(27,33,34,39)(28,36,38,37)(32,35,40,41)

Conjugacy classes & character table

952 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#16921195, #16921199, #16921201, …
maximal subgroup of	C₁₇×#7155, C₁₇×#7157, C₁₇×#7158, …
maximal normal subgroup of	C₁₇×#7155, C₁₇×#7157, C₁₇×#7158, …
maximal quotient of	C₁₇×#7872, C₄×C₁₇×#334, C₄×C₁₇×#335, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2176, 296);