C₂₃×#1515 #15669134

order	2944
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₂₃×#1515

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂³×C₂₃
prime radical	C₂³×C₂₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

208 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 88 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 — 3 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)(25,30,32,44)(26,35)(27,39)(31,46,47,55)(33,45,34,40)(37,52)(38,51)(41,42)(43,48,54,49)(50,53);(24,25,28,33)(26,31,38,48)(27,40,39,44)(29,34,36,32)(30,42,45,41)(35,49,51,47)(37,46,53,54)(43,52,55,50);(24,26,36,51)(25,31,32,47)(27,37,42,52)(28,38,29,35)(30,43,44,54)(33,48,34,49)(39,53,41,50)(40,46,45,55)

Conjugacy classes & character table

598 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₃×#8501, C₂₃×#8502, C₂₃×#8503, …
maximal normal subgroup of	C₂₃×#8501, C₂₃×#8502, C₂₃×#8503, …
maximal quotient of	C₂₃×#7596, C₂₃×#7604, C₂₃×#7605, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2944, 247);