C₃×#13091645 #72243626

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃⁴ · Schur multiplier (H₂) = C₃⁶

Aliases

direct product

C₃×#13091645

Splittings

Fitting, Sylow

C₃×#10624930 ⋊ C₂

Subgroups

	subgroup	G/H
center, socle	C₃²	C₃²×(C₃⋊D₃)⋊C₃
commutator, cosocle, hypocenter	H(3,𝔽₃)	C₂×C₃⁴
Frattini	C₃²	C₃⁴×D₃
Fitting	C₃×#10624930	C₂
hypercenter	C₃×H(3,𝔽₃)	(C₃⋊D₃)⋊C₃
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃² ◁ C₃×H(3,𝔽₃)

5347 conjugacy classes of subgroups; only the 668 normal subgroups are shown.

normal subgroups only

This group has 5347 conjugacy classes of subgroups — too many to draw. Only its 668 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) = 30
transitive degree	μ_t(G) = 81

A minimal faithful permutation representation — 4 generators on 30 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	#72243652, D₃×#13091645, C₃×#72221953, …
maximal normal subgroup of	#72243652, D₃×#13091645, C₃×#72221953, …

GAP

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

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