C₁₇×#1772 #14524778

order	2176
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₁₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₇×#1772

Subgroups

	subgroup	G/H
center	C₂×C₈×C₁₇	D₄
commutator	C₄	C₂×C₄²×C₁₇
Frattini, cosocle	C₂²×C₄	C₂³×C₁₇
socle	C₂²×C₁₇	C₄×D₄
prime radical	C₂³×C₁₇	#41
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₇ ◁ C₂²×C₈×C₁₇ ◁ G

232 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 152 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)(23,27,29,32)(28,34,35,36);(18,19,20,21)(22,23)(24,28)(25,32)(26,29)(27,33)(30,36)(31,35)(34,37);(22,24,25,30,26,31,33,37)(23,28,32,36,29,35,27,34)

Conjugacy classes & character table

952 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#16921264, #16921267, #16921270, …
maximal subgroup of	C₁₇×#7179, C₁₇×#7180, C₁₇×#7181, …
maximal normal subgroup of	C₁₇×#7179, C₁₇×#7180, C₁₇×#7181, …
maximal quotient of	C₁₇×#8266, C₁₇×#8267, C₁₇×#8270, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2176, 504);