D₁₁₉₃₅ #37558826

order	23870
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₉₃₅
unambiguous semidirect product	C₅⋊D₂₃₈₇
unambiguous semidirect product	C₇⋊D₁₇₀₅
unambiguous semidirect product	C₁₁⋊D₁₀₈₅
unambiguous semidirect product	C₃₁⋊D₃₈₅

Splittings

Fitting, socle, derived, Hall	C₅×C₇×C₁₁×C₃₁ ⋊ C₂
Hall	C₇×C₁₁×C₃₁ ⋊ D₅
Hall	C₅×C₁₁×C₃₁ ⋊ D₇
Hall	C₅×C₇×C₃₁ ⋊ D₁₁
Hall	C₅×C₇×C₁₁ ⋊ D₃₁
Hall	C₁₁×C₃₁ ⋊ D₃₅
Hall	C₇×C₃₁ ⋊ D₅₅
Hall	C₅×C₃₁ ⋊ D₇₇
Hall	C₇×C₁₁ ⋊ D₁₅₅
Hall	C₅×C₁₁ ⋊ D₂₁₇
Hall	C₅×C₇ ⋊ D₃₄₁
Sylow	C₃₁ ⋊ D₃₈₅
Sylow	C₁₁ ⋊ D₁₀₈₅
Sylow	C₇ ⋊ D₁₇₀₅
Sylow	C₅ ⋊ D₂₃₈₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₇×C₁₁×C₃₁	C₂
LCS	G ▷ C₅×C₇×C₁₁×C₃₁
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 54
transitive degree	μ_t(G) = 11935

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

Polycyclic presentation:

2,5,7,11,31|b^a=b^4,c^a=c^6,d^a=d^10,e^a=e^30

Conjugacy classes & character table

5969 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23870, 22);

D11935 #37558826