D₄×AGL(1,ℤ/27) #72228985

order	3888
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

D₄×AGL(1,ℤ/27)

Splittings

Fitting	D₄×#6875 ⋊ C₂
Sylow	#6875 ⋊ C₂×D₄

Subgroups

	subgroup	G/H
center	C₂	C₂²×AGL(1,ℤ/27)
commutator	C₂×C₂₇	C₂³×C₉
Frattini	C₂×C₃×C₉	C₂²×C₃×D₃
Fitting	D₄×#6875	C₂
socle	C₂×C₃	C₂²×D₉⋊C₉
prime radical	D₄×D₂₇⋊C₃	C₃
cosocle	C₂×C₂₇⋊C₃	C₂³×C₃
hypercenter	D₄	AGL(1,ℤ/27)
hypocenter	C₂₇	C₂×D₄×C₉
LCS	G ▷ C₂×C₂₇ ▷ C₂₇
UCS	C₁ ◁ C₂ ◁ D₄

356 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 93 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:

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

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	S₄×AGL(1,ℤ/27)
automorphism group of	Dic₅₄, D₁₀₈, C₂×Dic₂₇
holomorph of	C₄×C₂₇

GAP

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

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