C₁₇×#1967 #14524973

order	2176
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₇×#1967

Subgroups

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

312 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 112 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)(19,21)(22,23,26,29,25,28,32,34)(24,27,31,36,30,35,33,37);(22,24,26,33,25,30,32,31)(23,27,34,36,28,35,29,37);(18,19,20,21)(23,28)(27,35)(29,34)(36,37)

Conjugacy classes & character table

544 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#16926621, #16926628, #16938282, …
maximal subgroup of	C₁₇×#8483, C₁₇×#8484, C₁₇×#10496, …
maximal normal subgroup of	C₁₇×#8483, C₁₇×#8484, C₁₇×#10496, …
maximal quotient of	C₁₇×#9651, C₁₇×#9652, C₁₇×#9687, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2176, 699);