C₁₇×#1991 #14524997

order	2176
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₇×#1991

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₁₇
Frattini, cosocle	C₂²×C₄	C₂³×C₁₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

252 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 128 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) = 45
transitive degree	μ_t(G) = 2176

A minimal faithful permutation representation — 3 generators on 45 points:

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

Conjugacy classes & character table

748 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#16921150, #16921717, #16926873, …
maximal subgroup of	C₁₇×#7135, C₁₇×#7359, C₁₇×#8529, …
maximal normal subgroup of	C₁₇×#7135, C₁₇×#7359, C₁₇×#8529, …
maximal quotient of	C₁₇×#9871, C₁₇×#9875, C₁₇×#9886, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2176, 723);