C₈×D₂₇ #88358

order	432
isoclinism	isoclinic class of D₂₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₈×D₂₇

Splittings

Fitting	C₈×C₂₇ ⋊ C₂
derived, Sylow	C₂₇ ⋊ C₂×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₈	D₂₇
commutator, hypocenter	C₂₇	C₂×C₈
Frattini	C₄×C₉	C₂×D₃
Fitting	C₈×C₂₇	C₂
socle	C₂×C₃	C₄×D₉
prime radical	C₂×D₂₇	C₄
cosocle	C₄×C₂₇	C₂²
LCS	G ▷ C₂₇
UCS	C₁ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 216

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);(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)(28,29,30,32,31,33,34,35)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₁₆×D₂₇, #11720039, C₂×C₈×D₂₇, …
maximal normal subgroup of	C₁₆×D₂₇, #11720039, C₂×C₈×D₂₇, …
maximal quotient of	C₁₆×D₂₇, #11720039, C₈×Dic₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 5);

C8×D27 #88358

Aliases

Splittings

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₈×D₂₇ #88358