C₄×C₁₇×#126 #14525310

order	2176
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₁₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₄×C₁₇×#126

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₁₇	C₂³
commutator	C₂²	C₂³×C₄×C₁₇
Frattini, cosocle	C₂³	C₂⁴×C₁₇
socle	C₂³×C₁₇	C₂⁴
prime radical	C₂⁴×C₁₇	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₇ ◁ G

516 conjugacy classes of subgroups; only the 296 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 296 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) = 37
transitive degree	μ_t(G) = 1088

A minimal faithful permutation representation — 4 generators on 37 points:

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

Conjugacy classes & character table

952 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#126×#3622, #17042549, #17042560, …
maximal subgroup of	C₈×C₁₇×#126, C₁₇×#21665, C₁₇×#21666, …
maximal normal subgroup of	C₈×C₁₇×#126, C₁₇×#21665, C₁₇×#21666, …
maximal quotient of	C₄×C₁₇×#388, C₄×C₁₇×#387, C₄×C₁₇×#383, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2176, 1036);