C₅×D₁₂₃₅ #27134703

order	12350
automorphism	?
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×D₁₂₃₅
unambiguous semidirect product	C₅⋊(C₅×D₂₄₇)
unambiguous semidirect product	C₁₃⋊(C₅×D₉₅)
unambiguous semidirect product	C₁₉⋊(C₅×D₆₅)

Splittings

Fitting, socle, Hall	C₅²×C₁₃×C₁₉ ⋊ C₂
derived	C₅×C₁₃×C₁₉ ⋊ C₂×C₅
Hall	C₅²×C₁₉ ⋊ D₁₃
Hall	C₅²×C₁₃ ⋊ D₁₉
Hall	C₁₃×C₁₉ ⋊ C₅×D₅
Sylow	C₅² ⋊ D₂₄₇
Sylow	C₁₉ ⋊ C₅×D₆₅
Sylow	C₁₃ ⋊ C₅×D₉₅

Subgroups

	subgroup	G/H
center, hypercenter	C₅	D₁₂₃₅
commutator, cosocle, hypocenter	C₅×C₁₃×C₁₉	C₂×C₅
Frattini	C₁	G
Fitting, socle	C₅²×C₁₃×C₁₉	C₂
LCS	G ▷ C₅×C₁₃×C₁₉
UCS	C₁ ◁ C₅

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 42
transitive degree	μ_t(G) = 2470

A minimal faithful permutation representation — 2 generators on 42 points:

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

Conjugacy classes & character table

3095 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅×D₁₉×D₆₅, C₅×D₁₃×D₉₅, C₅×D₅×D₂₄₇, …
maximal normal subgroup of	C₅×D₁₉×D₆₅, C₅×D₁₃×D₉₅, C₅×D₅×D₂₄₇, …
maximal quotient of	C₅×Dic₁₂₃₅, C₂×C₅×D₁₂₃₅, C₃×C₅×D₁₂₃₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12350, 14);

C5×D1235 #27134703