C₂×D₂₉₇ #11924639

order	1188
isoclinism	isoclinic class of D₂₉₇
automorphism	Aut(G) order 106920
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₅₉₄
direct product	C₂×D₂₉₇
unambiguous semidirect product	(C₂×C₁₁)⋊D₂₇
unambiguous semidirect product	(C₂×C₂₇)⋊D₁₁

Splittings

Fitting	C₂×C₁₁×C₂₇ ⋊ C₂
derived, Hall	C₁₁×C₂₇ ⋊ C₂²
Sylow	C₂₇ ⋊ C₂×D₁₁
Sylow	C₁₁ ⋊ C₂×D₂₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₂₉₇
commutator, cosocle, hypocenter	C₁₁×C₂₇	C₂²
Frattini	C₉	C₂×D₃₃
Fitting	C₂×C₁₁×C₂₇	C₂
socle	C₂×C₃×C₁₁	D₉
LCS	G ▷ C₁₁×C₂₇
UCS	C₁ ◁ C₂

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 594

A minimal faithful permutation representation — 2 generators on 40 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);(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(31,40)(32,39)(33,38)(34,37)(35,36)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₁₁₈₈, C₂×D₁₄₈₅, C₂×D₈₉₁, …
maximal normal subgroup of	D₁₁₈₈
maximal quotient of	D₁₁₈₈, Dic₅₉₄, C₂×D₁₄₈₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1188, 9);

C2×D297 #11924639