C₇×C₂₇⋊C₈ #13094106

order	1512
isoclinism	isoclinic class of D₂₇
automorphism	?
homology	Abelianization (H₁) = C₇×C₈ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×C₂₇⋊C₈
unambiguous semidirect product	C₂₇⋊(C₇×C₈)

Splittings

derived, Sylow	C₂₇ ⋊ C₇×C₈
Hall	C₇×C₂₇ ⋊ C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₄×C₇	D₂₇
commutator, hypocenter	C₂₇	C₇×C₈
Frattini	C₄×C₉	D₃×C₇
Fitting	C₄×C₇×C₂₇	C₂
socle, prime radical	C₂×C₃×C₇	Dic₉
cosocle	C₄×C₂₇	C₂×C₇
LCS	G ▷ C₂₇
UCS	C₁ ◁ C₄×C₇

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 42
transitive degree	μ_t(G) = 1512

A minimal faithful permutation representation — 2 generators on 42 points:

(1,2,3,4,5,6,7)(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)(35,36,37,39,38,40,41,42);(8,27,21,28,32,30,10,9,31,11,34,25,17,14,13,18,15,33,19,23,12,16,24,22,29,26,20)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1512, 3);

C7×C27⋊C8 #13094106