Dic₇×C₄₃ #11925837

order	1204
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₄×C₄₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	Dic₇×C₄₃
unambiguous semidirect product	C₇⋊(C₄×C₄₃)

Splittings

derived, Sylow	C₇ ⋊ C₄×C₄₃
Hall	C₇×C₄₃ ⋊ C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄₃	D₇
commutator, hypocenter	C₇	C₄×C₄₃
Frattini	C₂	D₇×C₄₃
Fitting, socle, prime radical	C₂×C₇×C₄₃	C₂
cosocle	C₂×C₇	C₂×C₄₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₄₃

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 54
transitive degree	μ_t(G) = 1204

A minimal faithful permutation representation — 2 generators on 54 points:

(1,2)(3,4)(5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54);(1,3,2,4)(6,11)(7,10)(8,9)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	Dic₇×C₄₃⋊C₃, C₄₃×ΓL(1,𝔽₈)⋊C₄, #16813404, …
maximal normal subgroup of	Dic₇×C₄₃⋊C₃, C₄₃×ΓL(1,𝔽₈)⋊C₄, #16813404, …
maximal quotient of	C₃×Dic₇×C₄₃, C₄₃×Dic₂₁, C₄₃×C₇⋊#157, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1204, 3);

Dic7×C43 #11925837