C₇×D₁₀₈ #13094134

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

Aliases

direct product	C₇×D₁₀₈
unambiguous semidirect product	C₄⋊(C₇×D₂₇)

Splittings

Fitting	C₄×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	C₄×C₇×C₂₇	C₂
socle	C₂×C₃×C₇	C₂×D₉
hypercenter	C₄×C₇	D₂₇
hypocenter	C₂₇	C₇×D₄
LCS	G ▷ C₂×C₂₇ ▷ C₂₇
UCS	C₁ ◁ C₂×C₇ ◁ C₄×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)(8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38);(9,11)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)

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, 31);

C7×D108 #13094134