C₁₇×#13051 #16870795

order	4352
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₇×#13051

Subgroups

	subgroup	G/H
center	C₄×C₁₇	B(4,𝔽₂)
commutator	C₂×D₄	C₂²×C₄×C₁₇
Frattini, cosocle	C₂×#41	C₂³×C₁₇
socle	C₂×C₁₇	#1889
prime radical	C₁₇×#3018	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₇ ◁ C₂×C₄×C₁₇ ◁ C₂×#41×C₁₇ ◁ G

632 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 114 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)(18,19,23,28,21,26,30,39)(20,29,33,45,31,24,38,42)(22,27,36,44,34,25,37,43)(32,40,46,48,35,41,47,49);(18,20,30,38,21,31,23,33)(19,24,39,45,26,29,28,42)(22,35,36,47,34,32,37,46)(25,40,43,48,27,41,44,49);(19,25)(22,34)(24,40)(26,27)(28,44)(29,41)(32,35)(36,37)(39,43)(42,49)(45,48)(46,47)

Conjugacy classes & character table

680 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4352, 6039);