C₂₃×#554 #13092103

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

Aliases

direct product

C₂₃×#554

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

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

normal subgroups only

This group has 228 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) = 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)(24,25,28,27)(26,30,31,29)(32,33,36,35)(34,38,39,37);(25,27)(29,30)(32,34,36,39)(33,37,35,38);(24,26)(25,29)(27,30)(28,31);(24,27,28,25)(26,30,31,29)(32,35,36,33)(34,37,39,38)

Conjugacy classes & character table

575 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#15672469, #15672531, #15672604, …
maximal subgroup of	C₂₃×#1622, C₂₃×#1632, C₂₃×#1643, …
maximal normal subgroup of	C₂₃×#1622, C₂₃×#1632, C₂₃×#1643, …
maximal quotient of	C₂₃×#2345, D₄×C₄⋊C₄×C₂₃, C₂₃×#2349, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1472, 235);