C₂²×#21439061 #44476725

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

Aliases

direct product

C₂²×#21439061

Splittings

Fitting, socle, Hall	C₂²×C₇×C₁₉² ⋊ C₃
derived, Hall	C₇×C₁₉² ⋊ C₂²×C₃
Hall	C₂²×C₁₉² ⋊ ΓL(1,𝔽₈)
Sylow	C₁₉² ⋊ C₂²×ΓL(1,𝔽₈)
Hall	C₂²×C₇ ⋊ #11762245
Sylow	C₇ ⋊ C₂²×#11762245

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	#21439061
commutator, cosocle, hypocenter	C₇×C₁₉²	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂²×C₇×C₁₉²	C₃
LCS	G ▷ C₇×C₁₉²
UCS	C₁ ◁ C₂²

440 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 49
transitive degree	μ_t(G) = 10108

A minimal faithful permutation representation — 3 generators on 49 points:

(1,2)(6,7,8)(9,11,10)(13,14,16)(15,18,22)(17,20,19)(21,25,26)(23,27,24)(28,30,29)(32,33,35)(34,37,41)(36,39,38)(40,44,45)(42,46,43)(47,49,48);(3,4)(5,6,7,9,8,10,11)(31,32,34,37,42,47,46,35,38,43,44,33,36,40,41,45,48,49,39);(12,13,15,18,23,28,27,16,19,24,25,14,17,21,22,26,29,30,20)

Conjugacy classes & character table

3380 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30324, 308);

C22×#21439061 #44476725