order	304
isoclinism	isoclinic class of D₁₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₁₉⋊C₈
unambiguous semidirect product	(C₂×C₁₉)⋊C₈

Splittings

derived, Sylow

C₁₉ ⋊ C₂×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄	D₁₉
commutator, hypocenter	C₁₉	C₂×C₈
Frattini	C₄	C₂×D₁₉
Fitting	C₂×C₄×C₁₉	C₂
socle, prime radical	C₂²×C₁₉	C₄
cosocle	C₄×C₁₉	C₂²
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₂×C₄

22 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 304

A minimal faithful permutation representation — 2 generators on 29 points:

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×C₁₉⋊C₈, #10597260, C₄⋊(C₁₉⋊C₈), …
maximal normal subgroup of	C₄×C₁₉⋊C₈, #10597260, C₄⋊(C₁₉⋊C₈), …
maximal quotient of	C₄×C₁₉⋊C₈, C₄⋊(C₁₉⋊C₈), C₂×C₁₉⋊C₁₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(304, 8);

C2×C19⋊C8 #64519

Aliases

Splittings

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×C₁₉⋊C₈ #64519