C₄×Dic₂₇ #88364

order	432
isoclinism	isoclinic class of D₂₇
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₄

Aliases

direct product	C₄×Dic₂₇
unambiguous semidirect product	C₂₇⋊C₄²

Splittings

derived, Sylow

C₂₇ ⋊ C₄²

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄	D₂₇
commutator, hypocenter	C₂₇	C₄²
Frattini	C₂²×C₉	C₂×D₃
Fitting	C₂×C₄×C₂₇	C₂
socle, prime radical	C₂²×C₃	C₂×D₉
cosocle	C₂²×C₂₇	C₂²
LCS	G ▷ C₂₇
UCS	C₁ ◁ C₂×C₄

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 432

A minimal faithful permutation representation — 2 generators on 35 points:

(2,5)(3,9)(4,12)(6,7)(8,18)(10,11)(13,20)(14,19)(15,25)(16,24)(17,23)(21,22)(26,27)(32,33,34,35);(1,20,14,21,25,23,3,2,24,4,27,18,10,7,6,11,8,26,12,16,5,9,17,15,22,19,13)(28,29,30,31)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₈×Dic₂₇, Dic₂₇⋊C₈, #11720056, …
maximal normal subgroup of	C₈×Dic₂₇, Dic₂₇⋊C₈, #11720056, …
maximal quotient of	C₄×C₂₇⋊C₈, #11720043, C₈×Dic₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 11);

C4×Dic27 #88364

Aliases

Splittings

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₄×Dic₂₇ #88364