C₇×#6171 #13094136

order	1512
isoclinism	isoclinic class of Dic₅₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₂

supersolvable
metabelian
solvable
monomial

Aliases

direct product

C₇×#6171

Splittings

Fitting	C₂²×C₇×C₂₇ ⋊ C₂
Hall	C₇×C₂₇ ⋊ D₄
Sylow	C₂₇ ⋊ C₇×D₄

Subgroups

	subgroup	G/H
center	C₂×C₇	C₂×D₂₇
commutator, cosocle	C₂×C₂₇	C₂²×C₇
Frattini	C₂×C₉	C₂×D₃×C₇
Fitting	C₂²×C₇×C₂₇	C₂
socle	C₂×C₃×C₇	C₂×D₉
hypercenter	C₂²×C₇	D₂₇
hypocenter	C₂₇	C₇×D₄
LCS	G ▷ C₂×C₂₇ ▷ C₂₇
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₇

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 756

A minimal faithful permutation representation — 2 generators on 38 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1512, 33);

C7×#6171 #13094136