C₁₇×#1506 #14524512

order	2176
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₇×#1506

Subgroups

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

228 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 544

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)(19,22,23,24)(26,27)(28,34,36,31)(29,32)(30,33)(35,40,37,39)(38,41);(18,19)(20,24)(21,23)(22,25)(26,28,30,36)(27,31,33,34)(29,37,38,35)(32,40,41,39);(26,29,30,38)(27,32,33,41)(28,35,36,37)(31,39,34,40)

Conjugacy classes & character table

544 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#16926184, #16926198, #16926681, …
maximal subgroup of	C₁₇×#8413, C₁₇×#8415, C₁₇×#8492, …
maximal normal subgroup of	C₁₇×#8413, C₁₇×#8415, C₁₇×#8492, …
maximal quotient of	C₁₇×#7581, C₁₇×#7588, C₁₇×#7590, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2176, 238);