C₁₇×#1814 #14524820

order	2176
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₇×#1814

Subgroups

	subgroup	G/H
center	C₄×C₁₇	#121
commutator	C₂×C₄	C₂²×C₄×C₁₇
Frattini, cosocle	C₂²×C₄	C₂³×C₁₇
socle	C₂×C₁₇	#380
prime radical	C₂×#41×C₁₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₇ ◁ C₂²×C₄×C₁₇ ◁ G

256 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 116 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)(21,34,36,41)(22,37)(23,27)(24,28)(25,31)(26,35,42,38)(30,39,46,49)(32,40)(33,44)(43,45,48,47);(18,20,22,31,24,33,23,32)(19,25,27,29,28,40,37,44)(21,30,35,45,36,46,38,47)(26,39,41,43,42,49,34,48);(18,21)(19,26)(20,30)(22,35)(23,38)(24,36)(25,39)(27,41)(28,42)(29,43)(31,45)(32,47)(33,46)(34,37)(40,49)(44,48)

Conjugacy classes & character table

544 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#16950039, #16950060, #16950067, …
maximal subgroup of	C₁₇×#12747, C₁₇×#12750, C₁₇×#12751, …
maximal normal subgroup of	C₁₇×#12747, C₁₇×#12750, C₁₇×#12751, …
maximal quotient of	C₁₇×#8604, C₁₇×#8605, C₁₇×#8606, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2176, 546);