C₇×D₃₃² #44479209

order	30492
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×D₃₃²

Splittings

Fitting, socle, Hall	C₃²×C₇×C₁₁² ⋊ C₂²
derived, Hall	C₃²×C₁₁² ⋊ C₂²×C₇
Hall	C₇×C₁₁² ⋊ D₃²
Sylow	C₁₁² ⋊ D₃²×C₇
Hall	C₃²×C₇ ⋊ D₁₁²
Sylow	C₃² ⋊ C₇×D₁₁²

Subgroups

	subgroup	G/H
center, hypercenter	C₇	D₃₃²
commutator, cosocle, hypocenter	C₃²×C₁₁²	C₂²×C₇
Frattini	C₁	G
Fitting, socle	C₃²×C₇×C₁₁²	C₂²
LCS	G ▷ C₃²×C₁₁²
UCS	C₁ ◁ C₇

296 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 462

A minimal faithful permutation representation — 2 generators on 35 points:

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

Conjugacy classes & character table

2268 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30492, 432);