C₁₇×#1824 #14524830

order	2176
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₇×#1824

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₁₇
Frattini, cosocle	C₂²×C₄	C₂³×C₁₇
prime radical	C₂⁴×C₁₇	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ C₂³×C₄×C₁₇ ◁ G

344 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 1088

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

Conjugacy classes & character table

646 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#16926001, #16926011, #16929350, …
maximal subgroup of	C₁₇×#8378, C₁₇×#8380, C₁₇×#9021, …
maximal normal subgroup of	C₁₇×#8378, C₁₇×#8380, C₁₇×#9021, …
maximal quotient of	C₁₇×#8707, C₁₇×#8710, C₁₇×#8713, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2176, 556);