C₄×#3026⋊C₃ #72245656

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₄×#3026⋊C₃

Splittings

Fitting, Sylow

C₄×#3026 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120⋊C₃
commutator, hypocenter	H(5,𝔽₂)	C₂²×C₃×C₄
Frattini	C₂⁴	C₂³×A₄
Fitting	C₄×#3026	C₃
socle	C₂²	C₂²×#120⋊C₃
prime radical	C₂×#3026⋊C₃	C₂
cosocle	C₂×H(5,𝔽₂)	C₂³×C₃
hypercenter	C₂²×C₄	#120⋊C₃
LCS	G ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

1957 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 1957 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 20 points:

(6,10)(7,14,8)(11,19,12,13,18,16)(15,20,17);(1,2,3,4)(5,14)(6,18)(7,8)(9,20)(10,19)(11,12)(13,16)(15,17);(5,6)(7,13)(8,16)(9,10)(11,15)(12,17)(14,19)(18,20)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#72209831, C₄×#11714170, C₄×#11714182, …
maximal normal subgroup of	C₄×#11714170, C₄×#11714182, C₄×#60399⋊C₃

GAP

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

gap> G := Group((6,10)(7,14,8)(11,19,12,13,18,16)(15,20,17), (1,2,3,4)(5,14)(6,18)(7,8)(9,20)(10,19)(11,12)(13,16)(15,17), (5,6)(7,13)(8,16)(9,10)(11,15)(12,17)(14,19)(18,20));