C₂²×#3549⋊C₃ #72337539

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

direct product

C₂²×#3549⋊C₃

Splittings

Fitting, Sylow	C₂²×#3549 ⋊ C₃
derived	H(3,𝔽₄) ⋊ C₂³×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	#3549⋊C₃
commutator, cosocle, hypocenter	H(3,𝔽₄)	C₂³×C₃
Frattini	C₂²	C₂³×C₂²⋊A₄
Fitting	C₂²×#3549	C₃
socle	C₂⁴	C₂×C₂²⋊A₄
LCS	G ▷ H(3,𝔽₄)
UCS	C₁ ◁ C₂²

14262 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 14262 conjugacy classes of subgroups — too many to draw. Only its 133 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) = 26
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 3 generators on 26 points:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#11715754, C₂²×#63011⋊C₃, C₂²×#11714278, …
maximal normal subgroup of	C₂²×#11715754, C₂²×#63011⋊C₃, C₂²×#11714278, …
maximal quotient of	C₂×#72237719, C₂×C₄×#3549⋊C₃

GAP

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

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