C₁₇×#1759 #14524765

order	2176
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₁₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₇×#1759

Subgroups

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

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

normal subgroups only

This group has 320 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)(18,19,21,20)(22,23)(24,30)(25,33)(26,29)(27,32)(28,34)(31,36)(35,37);(22,24)(23,27,29,35)(25,32)(26,31)(28,30,33,36)(34,37);(18,20,21,19)(24,31)(27,35)(30,36)(32,37)

Conjugacy classes & character table

748 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#16926176, #16926183, #16926190, …
maximal subgroup of	C₁₇×#8412, C₁₇×#8413, C₁₇×#8414, …
maximal normal subgroup of	C₁₇×#8412, C₁₇×#8413, C₁₇×#8414, …
maximal quotient of	C₁₇×#8170, C₁₇×#8171, C₁₇×#8174, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2176, 491);