C₇×Dic₅₄ #13094132

order	1512
isoclinism	isoclinic class of Dic₅₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×Dic₅₄
unambiguous semidirect product	C₂₇⋊(C₇×Q₈)

Splittings

Hall	C₇×C₂₇ ⋊ Q₈
Sylow	C₂₇ ⋊ C₇×Q₈

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, prime radical	C₂×C₃×C₇	C₂×D₉
hypercenter	C₄×C₇	D₂₇
hypocenter	C₂₇	C₇×Q₈
LCS	G ▷ C₂×C₂₇ ▷ C₂₇
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇

48 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)(8,9,10,11)(12,15,14,13)(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42);(8,12,10,14)(9,13,11,15)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)

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, 29);

C7×Dic54 #13094132