C₂²⋊C₄×D₂₇⋊C₃ #72276208

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂²⋊C₄×D₂₇⋊C₃

Splittings

Fitting	C₂²⋊C₄×C₂₇⋊C₃ ⋊ C₂
Sylow	C₂₇⋊C₃ ⋊ C₂×C₂²⋊C₄

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₂₇⋊C₃
commutator	C₂×C₂₇	C₂²×C₃×C₄
Frattini	C₂²×C₉	C₂²×C₃×D₃
Fitting	C₂²⋊C₄×C₂₇⋊C₃	C₂
socle	C₂²×C₃	C₂²×C₃×D₉
prime radical	C₂³×D₂₇⋊C₃	C₂
cosocle	C₂²×C₂₇	C₂³×C₃
hypercenter	C₂²⋊C₄	D₂₇⋊C₃
hypocenter	C₂₇	C₂×C₃×C₂²⋊C₄
LCS	G ▷ C₂×C₂₇ ▷ C₂₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

562 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 131 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) = 35
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 3 generators on 35 points:

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

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

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