C₂₃×#548 #13092097

order	1472
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₂₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂₃×#548

Subgroups

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

290 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 35
transitive degree	μ_t(G) = 736

A minimal faithful permutation representation — 4 generators on 35 points:

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

Conjugacy classes & character table

575 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#15684893, #15684953, #15684963, …
maximal subgroup of	C₂₃×#3265, C₂₃×#3269, C₂₃×#3270, …
maximal normal subgroup of	C₂₃×#3265, C₂₃×#3269, C₂₃×#3270, …
maximal quotient of	Q₈×C₂²⋊C₄×C₂₃, C₂₃×#2341, C₂₃×#2351, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1472, 229);