D₄×#4183⋊C₃ #72228564

order	3888
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

D₄×#4183⋊C₃

Splittings

Fitting	D₄×#6909 ⋊ C₂
Sylow	#6909 ⋊ C₂×D₄

Subgroups

	subgroup	G/H
center	C₂	C₂²×#4183⋊C₃
commutator, cosocle	C₂×AΣL(1,𝔽₂₇)	C₂³×C₃
Frattini	C₂×C₃²	C₂²×C₃×C₃⋊D₃
Fitting	D₄×#6909	C₂
socle	C₂×C₃	C₂²×C₃×#272
hypercenter	D₄	#4183⋊C₃
hypocenter	AΣL(1,𝔽₂₇)	C₂×C₃×D₄
LCS	G ▷ C₂×AΣL(1,𝔽₂₇) ▷ AΣL(1,𝔽₂₇)
UCS	C₁ ◁ C₂ ◁ D₄

1262 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 1262 conjugacy classes of subgroups — too many to draw. Only its 128 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) = 31
transitive degree	μ_t(G) = 108

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

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

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

D₄×#13091120

GAP

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

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