C₁₇×#1517 #14524523

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

Aliases

direct product

C₁₇×#1517

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	C₁₇×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

264 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 104 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)(18,19)(20,24)(21,23)(22,25)(26,27)(28,34,36,31)(29,32)(30,33)(35,40,37,39)(38,41);(19,23)(20,21)(24,25)(26,28)(27,31)(29,37)(30,36)(32,40)(33,34)(35,38)(39,41);(18,20,22,21)(19,23,25,24)(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	#16926289, #16926291, #16926296, …
maximal subgroup of	C₁₇×#8428, C₁₇×#8429, C₁₇×#8430, …
maximal normal subgroup of	C₁₇×#8428, C₁₇×#8429, C₁₇×#8430, …
maximal quotient of	C₁₇×#7574, C₁₇×#7587, C₁₇×#7588, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2176, 249);