C₄×#21439061 #44476538

order	30324
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₄×#21439061

Splittings

Fitting, Hall	C₄×C₇×C₁₉² ⋊ C₃
derived, Hall	C₇×C₁₉² ⋊ C₃×C₄
Hall	C₄×C₁₉² ⋊ ΓL(1,𝔽₈)
Sylow	C₁₉² ⋊ C₄×ΓL(1,𝔽₈)
Hall	C₄×C₇ ⋊ #11762245
Sylow	C₇ ⋊ C₄×#11762245

Subgroups

	subgroup	G/H
center, hypercenter	C₄	#21439061
commutator, hypocenter	C₇×C₁₉²	C₃×C₄
Frattini	C₂	C₂×#21439061
Fitting	C₄×C₇×C₁₉²	C₃
socle, cosocle	C₂×C₇×C₁₉²	C₂×C₃
prime radical	C₂×#21439061	C₂
LCS	G ▷ C₇×C₁₉²
UCS	C₁ ◁ C₄

264 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:

(1,2,3,4)(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);(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, 121);

C4×#21439061 #44476538