Finite Group Databasebeta

← Prev (#90683) Next (#90685) →

C₃×C₁₉⋊C₈ #90684

order	456
isoclinism	isoclinic class of D₁₉
automorphism	?
homology	Abelianization (H₁) = C₃×C₈ · Schur multiplier (H₂) = C₁

Aliases

C₃×C₁₉⋊C₈

Splittings

derived, Sylow	C₁₉ ⋊ C₃×C₈
Hall	C₃×C₁₉ ⋊ C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₄	D₁₉
commutator, hypocenter	C₁₉	C₃×C₈
Frattini	C₄	C₃×D₁₉
Fitting	C₃×C₄×C₁₉	C₂
socle, prime radical	C₂×C₃×C₁₉	C₄
cosocle	C₄×C₁₉	C₂×C₃
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₃×C₄

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 456

A minimal faithful permutation representation — 2 generators on 30 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,24,26,21,23,27,22,25)(28,29,30);(1,2,4,6,8,10,12,14,16,18,19,17,15,13,11,9,7,5,3)

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×C₁₉⋊C₈, D₅₇⋊C₈, #11744816, …
maximal normal subgroup of	D₃×C₁₉⋊C₈, D₅₇⋊C₈, #11744816, …
maximal quotient of	C₃×C₁₉⋊C₁₆, C₂×C₃×C₁₉⋊C₈, C₉×C₁₉⋊C₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(456, 4);