C₁₇×#12723 #16870467

order	4352
nilpotency class	4
isoclinism	isoclinic class of #12721
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₇×#12723

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₇	#1782
commutator	C₂²×C₄	C₂²×C₄×C₁₇
Frattini, cosocle	C₂×#41	C₂³×C₁₇
prime radical	C₁₇×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₇ ◁ C₂×C₄×C₁₇ ◁ C₄×#41×C₁₇ ◁ G

694 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 49
transitive degree	μ_t(G) = 544

A minimal faithful permutation representation — 3 generators on 49 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,29,32,44)(21,26,34,42)(22,27)(23,28)(24,30)(25,33,41,48)(31,37,35,49)(36,40)(38,45)(39,46)(43,47);(18,20,23,32)(19,24,28,39)(21,30,34,46)(22,35,36,31)(25,37,41,49)(26,29,42,44)(27,43,40,38)(33,47,48,45);(18,21,23,34)(19,25)(20,30,32,46)(22,33,36,48)(24,37)(26,27)(28,41)(29,43)(31,45,35,47)(38,44)(39,49)(40,42)

Conjugacy classes & character table

578 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4352, 5711);