C₇×Dic₄₃ #11925838

order	1204
isoclinism	isoclinic class of D₄₃
automorphism	?
homology	Abelianization (H₁) = C₄×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₇×Dic₄₃

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₇	D₄₃
commutator, hypocenter	C₄₃	C₄×C₇
Frattini	C₂	C₇×D₄₃
Fitting, socle, prime radical	C₂×C₇×C₄₃	C₂
cosocle	C₂×C₄₃	C₂×C₇
LCS	G ▷ C₄₃
UCS	C₁ ◁ C₂×C₇

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₇×(C₄₃⋊C₃)⋊C₄, ΓL(1,𝔽₈)×Dic₄₃, #16813400, …
maximal normal subgroup of	C₇×(C₄₃⋊C₃)⋊C₄, ΓL(1,𝔽₈)×Dic₄₃, #16813400, …
maximal quotient of	C₃×C₇×Dic₄₃, C₇×Dic₁₂₉, C₇×C₄₃⋊#157, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1204, 4);

C7×Dic43 #11925838