C₂×D₃₀₁ #11925846

order	1204
isoclinism	isoclinic class of D₃₀₁
automorphism	Aut(G) order 151704
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₆₀₂
direct product	C₂×D₃₀₁
unambiguous semidirect product	(C₂×C₇)⋊D₄₃
unambiguous semidirect product	(C₂×C₄₃)⋊D₇

Splittings

Fitting, socle	C₂×C₇×C₄₃ ⋊ C₂
derived, Hall	C₇×C₄₃ ⋊ C₂²
Sylow	C₄₃ ⋊ C₂×D₇
Sylow	C₇ ⋊ C₂×D₄₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃₀₁
commutator, cosocle, hypocenter	C₇×C₄₃	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₄₃	C₂
LCS	G ▷ C₇×C₄₃
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 602

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#14268594, C₂×#14268593, C₂×ΓL(1,𝔽₈)⋊D₄₃, …
maximal normal subgroup of	C₂×#14268594, C₂×#14268593, C₂×ΓL(1,𝔽₈)⋊D₄₃, …
maximal quotient of	D₃×D₃₀₁, C₂×C₃×D₃₀₁, C₂×D₉₀₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1204, 12);

C2×D301 #11925846