C₂×#11923454 #72226001

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

solvable

Aliases

direct product

C₂×#11923454

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

1497 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 1497 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)(3,41,19)(5,6,46)(7,39,27)(8,44,21)(9,43,18)(10,17,47)(12,35,34)(13,26,29)(14,22,15)(16,37,50)(20,24,48)(23,40,25)(28,31,32)(30,38,45);(4,11)(5,17)(6,8)(7,24)(9,23)(12,30)(14,26)(15,29)(16,28)(18,43)(19,46)(20,48)(21,41)(25,40)(27,39)(31,36)(32,50)(33,37)(34,45)(35,42)(38,49)(44,47);(3,4)(5,16)(6,22)(7,15)(8,13)(9,26)(10,11)(12,17)(14,24)(18,42)(19,45)(20,38)(21,30)(23,29)(25,37)(27,36)(28,41)(31,43)(32,46)(33,48)(34,44)(35,39)(40,49)(47,50)

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226236, C₂×#72215280, D₃×#11923454
maximal normal subgroup of	#72226236, C₂×#72215280
maximal quotient of	#72226236, D₃×#11923454

GAP

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

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