C₁₇×#12724 #16870468

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×#12724

Subgroups

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

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

normal subgroups only

This group has 670 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)(21,33,35,26)(22,27)(23,28)(24,30,40,47)(25,34,41,49)(31,44)(32,45)(36,38)(37,42)(39,46,43,48);(18,20,23,32)(19,24,28,40)(21,30,35,47)(22,36,37,31)(25,38,41,44)(26,29,33,45)(27,43,42,39)(34,48,49,46);(18,21,23,35)(19,25)(20,30,32,47)(22,34,37,49)(24,38)(26,42)(27,33)(28,41)(29,39)(31,46,36,48)(40,44)(43,45)

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, 5712);