C₂×#14512506 #72277362

order	3840
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

solvable

Aliases

direct product

C₂×#14512506

Splittings

Fitting	C₂×C₅×H(5,𝔽₂) ⋊ Dic₃
Sylow	C₅ ⋊ C₂×#73153

Subgroups

	subgroup	G/H
center	C₂²	#11756729
commutator	C₅×#41⋊C₃	C₂²×C₄
Frattini	C₄	C₂²×A₄⋊AGL(1,𝔽₅)
Fitting	C₂×C₅×H(5,𝔽₂)	Dic₃
socle	C₂²×C₅	#5432
prime radical	C₂×D₅×#994	C₂
cosocle	D₅×#41⋊C₃	C₂³
hypercenter	C₂×Q₈	A₄⋊AGL(1,𝔽₅)
hypocenter	C₅×SL(2,𝔽₃)	C₂×C₂²⋊C₄
LCS	G ▷ C₅×#41⋊C₃ ▷ C₅×SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₈

2318 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 3 generators on 31 points:

(3,4,7,5,6)(8,25,24,9,26,29,14,22,31,13,18,19)(10,20,23,17,11,28,16,15,30,12,21,27);(1,2);(4,5,6,7)(9,12)(10,16)(11,19)(13,17)(15,23)(18,27)(20,30)(21,29)(22,24)(25,31)(26,28)

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

C₂×#72229366

GAP

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

gap> G := Group((3,4,7,5,6)(8,25,24,9,26,29,14,22,31,13,18,19)(10,20,23,17,11,28,16,15,30,12,21,27), (1,2), (4,5,6,7)(9,12)(10,16)(11,19)(13,17)(15,23)(18,27)(20,30)(21,29)(22,24)(25,31)(26,28));

C2×#14512506 #72277362