C₂₃×#556 #13092105

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

Aliases

direct product

C₂₃×#556

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₂₃
prime radical	C₂×D₄×C₂₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ G

212 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 39
transitive degree	μ_t(G) = 736

A minimal faithful permutation representation — 4 generators on 39 points:

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

Conjugacy classes & character table

575 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#15672507, #15672569, #15672653, …
maximal subgroup of	C₂₃×#1628, C₂₃×#1638, C₂₃×#1650, …
maximal normal subgroup of	C₂₃×#1628, C₂₃×#1638, C₂₃×#1650, …
maximal quotient of	C₂₃×#2347, C₂₃×#2354, C₂₃×#2355, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1472, 237);