C₂×#11923452 #72226004

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#11923452

Splittings

derived

Ω⁺(4,𝔽₃) ⋊ C₂³

Subgroups

	subgroup	G/H
center, socle, hypercenter	C₂²	S₄²
commutator, cosocle, hypocenter	Ω⁺(4,𝔽₃)	C₂³
Frattini	C₂	C₂×S₄²
Fitting	C₂×H(5,𝔽₂)	D₃²
LCS	G ▷ Ω⁺(4,𝔽₃)
UCS	C₁ ◁ C₂²

1502 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 1502 conjugacy classes of subgroups — too many to draw. Only its 47 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) = 50
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 50 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226254, C₂×#72215280, D₃×#11923452
maximal normal subgroup of	#72226254, C₂×#72215280
maximal quotient of	#72226254, D₃×#11923452

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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