C₂⁴×D₂₇⋊C₃ #72276255

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂⁴×D₂₇⋊C₃

Splittings

Fitting	C₂⁴×C₂₇⋊C₃ ⋊ C₂
derived	C₂₇ ⋊ C₂⁵×C₃
Sylow	C₂₇⋊C₃ ⋊ C₂⁵

Subgroups

	subgroup	G/H
center, hypercenter	C₂⁴	D₂₇⋊C₃
commutator, cosocle, hypocenter	C₂₇	C₂⁵×C₃
Frattini	C₉	C₂⁴×C₃×D₃
Fitting	C₂⁴×C₂₇⋊C₃	C₂
socle	C₂⁴×C₃	C₃×D₉
LCS	G ▷ C₂₇
UCS	C₁ ◁ C₂⁴

3126 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 35 points:

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

Conjugacy classes & character table

336 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	#120×D₂₇⋊C₃
maximal normal subgroup of	#120×D₂₇⋊C₃

GAP

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

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

C24×D27⋊C3 #72276255