C₁₇×#12772 #16870516

order	4352
nilpotency class	4
isoclinism	isoclinic class of #2202
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₇×#12772

Subgroups

	subgroup	G/H
center	C₄×C₁₇	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₁₇
Frattini, cosocle	C₂×#41	C₂³×C₁₇
socle	C₂×C₁₇	#1790
prime radical	C₁₇×#3529	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₇ ◁ C₂×C₄×C₁₇ ◁ C₂×#41×C₁₇ ◁ G

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

normal subgroups only

This group has 760 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)(20,30)(21,34,22,27)(23,28)(24,29)(25,31,41,36)(26,35,42,37)(32,39)(33,43)(38,45)(40,46,44,47)(48,49);(18,20)(19,25)(21,31)(22,36)(23,32)(24,33)(26,39)(27,43)(28,44)(29,41)(30,34)(35,46)(37,47)(38,48)(40,45)(42,49);(18,21)(19,26)(20,31)(22,24)(23,37)(25,39)(27,28)(29,42)(30,40)(32,47)(33,36)(34,45)(35,38)(41,49)(43,44)(46,48)

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, 5760);