C₄×#19097890 #34117402

order	20124
automorphism	?
homology	Abelianization (H₁) = C₄×C₉ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₄×#19097890

Splittings

derived, Hall	C₁₃×C₄₃ ⋊ C₄×C₉
Hall	C₄×C₁₃×C₄₃ ⋊ C₉
Hall	C₄×C₄₃ ⋊ C₁₃⋊C₉
Hall	C₄×C₁₃ ⋊ C₄₃⋊C₉
Sylow	C₄₃ ⋊ C₄×C₁₃⋊C₉
Sylow	C₁₃ ⋊ C₄×C₄₃⋊C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₄	#13131975
commutator, hypocenter	C₁₃×C₄₃	C₄×C₉
Frattini	C₂×C₃	C₂×#13131975
Fitting	C₃×C₄×C₁₃×C₄₃	C₃
socle, prime radical, cosocle	C₂×C₃×C₁₃×C₄₃	C₂×C₃
LCS	G ▷ C₁₃×C₄₃
UCS	C₁ ◁ C₃×C₄

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 20124

A minimal faithful permutation representation — 2 generators on 69 points:

(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,30)(18,26,31)(21,29,28)(23,32,25)(27,36,38)(33,39,37)(34,40,35)(41,43,42)(45,46,48)(47,50,53)(49,52,51)(54,56,55)(57,64,63,59,58,65,62,61,60)(66,67,68,69);(1,2,4,8,15,17,3,6,11,18,27,26,7,13,21,30,38,36,14,23,33,40,22,31,24,34,41,43,32,12,19,28,37,42,39,20,5,9,16,25,35,29,10)(44,45,47,46,49,53,50,54,56,48,51,55,52)

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(20124, 5);

C4×#19097890 #34117402