C₂×#29503114 #44476643

order	30324
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₂×#29503114

Splittings

Fitting, socle	C₂×C₇×C₁₉² ⋊ C₂×C₃
derived, Hall	C₇×C₁₉² ⋊ C₂²×C₃
Hall	C₂×D₇×C₁₉² ⋊ C₃
Hall	#21439062 ⋊ C₂²
Sylow	C₁₉² ⋊ C₂×AGL(1,𝔽₇)
Hall	C₂×D₇ ⋊ #11762245
Sylow	C₇ ⋊ C₂²×#11762245

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#29503114
commutator, cosocle, hypocenter	C₇×C₁₉²	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₁₉²	C₂×C₃
LCS	G ▷ C₇×C₁₉²
UCS	C₁ ◁ C₂

440 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 47
transitive degree	μ_t(G) = 5054

A minimal faithful permutation representation — 3 generators on 47 points:

(4,7,8,5,6,9)(11,12,14)(13,16,17)(15,19,18)(20,24,22)(21,25,23)(26,28,27)(30,31,33)(32,35,36)(34,38,37)(39,43,41)(40,44,42)(45,47,46);(1,2)(3,4,6,9,8,7,5)(29,30,32,36,41,46,43,31,34,39,44,33,37,42,35,40,45,47,38);(10,11,13,17,22,27,24,12,15,20,25,14,18,23,16,21,26,28,19)

Conjugacy classes & character table

1214 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30324, 226);

C2×#29503114 #44476643