D₁₃₆₈₅ #41032852

order	27370
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₃₆₈₅
unambiguous semidirect product	C₅⋊D₂₇₃₇
unambiguous semidirect product	C₇⋊D₁₉₅₅
unambiguous semidirect product	C₁₇⋊D₈₀₅
unambiguous semidirect product	C₂₃⋊D₅₉₅

Splittings

Fitting, socle, derived, Hall	C₅×C₇×C₁₇×C₂₃ ⋊ C₂
Hall	C₇×C₁₇×C₂₃ ⋊ D₅
Hall	C₅×C₁₇×C₂₃ ⋊ D₇
Hall	C₅×C₇×C₂₃ ⋊ D₁₇
Hall	C₅×C₇×C₁₇ ⋊ D₂₃
Hall	C₁₇×C₂₃ ⋊ D₃₅
Hall	C₇×C₂₃ ⋊ D₈₅
Hall	C₇×C₁₇ ⋊ D₁₁₅
Hall	C₅×C₂₃ ⋊ D₁₁₉
Hall	C₅×C₁₇ ⋊ D₁₆₁
Hall	C₅×C₇ ⋊ D₃₉₁
Sylow	C₂₃ ⋊ D₅₉₅
Sylow	C₁₇ ⋊ D₈₀₅
Sylow	C₇ ⋊ D₁₉₅₅
Sylow	C₅ ⋊ D₂₇₃₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₇×C₁₇×C₂₃	C₂
LCS	G ▷ C₅×C₇×C₁₇×C₂₃
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 13685

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

Polycyclic presentation:

2,5,7,17,23|b^a=b^4,c^a=c^6,d^a=d^16,e^a=e^22

Conjugacy classes & character table

6844 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₃₆₈₅
maximal normal subgroup of	C₂×D₁₃₆₈₅
maximal quotient of	Dic₁₃₆₈₅, C₂×D₁₃₆₈₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27370, 16);

D13685 #41032852