C₃×Dic₃₈₅ #17987614

order	4620
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×Dic₃₈₅
unambiguous semidirect product	C₁₁⋊(C₃×Dic₃₅)

Splittings

derived, Hall	C₅×C₇×C₁₁ ⋊ C₃×C₄
Hall	C₃×C₅×C₇×C₁₁ ⋊ C₄
Hall	C₃×C₇×C₁₁ ⋊ Dic₅
Hall	C₃×C₅×C₁₁ ⋊ Dic₇
Hall	C₃×C₅×C₇ ⋊ Dic₁₁
Hall	C₇×C₁₁ ⋊ C₃×Dic₅
Hall	C₅×C₁₁ ⋊ C₃×Dic₇
Hall	C₅×C₇ ⋊ C₃×Dic₁₁
Hall	C₃×C₁₁ ⋊ Dic₃₅
Hall	C₃×C₇ ⋊ Dic₅₅
Hall	C₃×C₅ ⋊ Dic₇₇
Sylow	C₁₁ ⋊ C₃×Dic₃₅
Sylow	C₇ ⋊ C₃×Dic₅₅
Sylow	C₅ ⋊ C₃×Dic₇₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	D₃₈₅
commutator, hypocenter	C₅×C₇×C₁₁	C₃×C₄
Frattini	C₂	C₃×D₃₈₅
Fitting, socle, prime radical	C₂×C₃×C₅×C₇×C₁₁	C₂
cosocle	C₂×C₅×C₇×C₁₁	C₂×C₃
LCS	G ▷ C₅×C₇×C₁₁
UCS	C₁ ◁ C₂×C₃

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 4620

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

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

Conjugacy classes & character table

1164 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₉×Dic₃₈₅, C₁₁⋊((C₇⋊C₉)⋊Dic₅), C₃×C₁₁⋊(ΓL(1,𝔽₈)⋊Dic₅), …
maximal normal subgroup of	C₉×Dic₃₈₅, C₁₁⋊((C₇⋊C₉)⋊Dic₅), C₃×C₁₁⋊(ΓL(1,𝔽₈)⋊Dic₅), …
maximal quotient of	C₉×Dic₃₈₅, C₃²×Dic₃₈₅, C₃×Dic₁₁₅₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4620, 11);

C3×Dic385 #17987614