D₆₁₇₅ #27134697

order	12350
automorphism	Aut(G) order 26676000
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₆₁₇₅
unambiguous semidirect product	C₁₃⋊D₄₇₅
unambiguous semidirect product	C₁₉⋊D₃₂₅
unambiguous semidirect product	C₂₅⋊D₂₄₇

Splittings

Fitting, derived, Hall	C₁₃×C₁₉×C₂₅ ⋊ C₂
Hall	C₁₉×C₂₅ ⋊ D₁₃
Hall	C₁₃×C₂₅ ⋊ D₁₉
Hall	C₁₃×C₁₉ ⋊ D₂₅
Sylow	C₂₅ ⋊ D₂₄₇
Sylow	C₁₉ ⋊ D₃₂₅
Sylow	C₁₃ ⋊ D₄₇₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₁₃×C₁₉×C₂₅	C₂
Frattini	C₅	D₁₂₃₅
socle	C₅×C₁₃×C₁₉	D₅
LCS	G ▷ C₁₃×C₁₉×C₂₅
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 6175

A minimal faithful permutation representation — 2 generators on 57 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)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57);(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)

Polycyclic presentation:

2,5,13,19,5|b^5=e^4,b^a=b^4e,c^a=c^12,d^a=d^18,e^a=e^4

Conjugacy classes & character table

3089 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₉×D₃₂₅, D₁₃×D₄₇₅, D₂₅×D₂₄₇, …
maximal normal subgroup of	D₁₉×D₃₂₅, D₁₃×D₄₇₅, D₂₅×D₂₄₇, …
maximal quotient of	Dic₆₁₇₅, C₂×D₆₁₇₅, C₃×D₆₁₇₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12350, 8);

D6175 #27134697