C₂₃×#1389 #15669008

order	2944
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄²×C₂₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂₃×#1389

Subgroups

	subgroup	G/H
center	C₄×C₂₃	#102
commutator	C₈	C₄²×C₂₃
Frattini, cosocle	C₂²×C₈	C₂²×C₂₃
socle	C₂×C₂₃	#340
prime radical	C₂³×C₂₃	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂²×C₈×C₂₃ ◁ G

128 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 55
transitive degree	μ_t(G) = 736

A minimal faithful permutation representation — 2 generators on 55 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23)(24,25,28,33,30,34,29,45)(26,35,37,52,40,48,39,54)(27,41,42,50,44,55,43,32)(31,46,47,38,49,53,51,36);(24,26,27,38,28,37,42,53,30,40,44,36,29,39,43,46)(25,31,32,48,45,51,55,52,34,49,50,35,33,47,41,54)

Conjugacy classes & character table

736 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₂₃×#1389, C₂₃×#12646, C₂₃×#12649, …
maximal normal subgroup of	C₂×C₂₃×#1389, C₂₃×#12646, C₂₃×#12649, …
maximal quotient of	C₂₃×#7159, C₂₃×#7175, C₂₃×#7177, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2944, 121);