C₅×#82028 #14478613

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#82028

Splittings

Fitting	C₂×C₃×C₅×Q₃₂ ⋊ C₂
Hall	C₃×C₅ ⋊ #3417
Sylow	C₃ ⋊ C₅×#3417

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂×D₃×D₈
commutator, cosocle	C₃×C₈	C₂⁴×C₅
Frattini	C₈	C₂³×C₅×D₃
Fitting	C₂×C₃×C₅×Q₃₂	C₂
socle	C₂×C₃×C₅	C₂²×D₈
hypercenter	C₂×C₅×Q₃₂	D₃
hypocenter	C₃	C₅×#3417
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₂×C₅×Q₃₂

700 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 960

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2,3,4,5)(6,7,8)(12,26)(17,40)(22,50)(27,54)(28,55)(37,63)(41,65)(42,59)(48,70)(51,62)(52,60)(56,69)(57,72)(58,66)(64,68)(67,71);(7,8)(9,12)(10,17)(11,22)(13,27)(14,28)(15,26)(16,37)(18,41)(19,42)(20,40)(21,48)(23,51)(24,52)(25,50)(29,56)(30,57)(31,55)(32,58)(33,54)(34,59)(35,60)(36,62)(38,64)(39,63)(43,66)(44,67)(45,65)(46,68)(47,69)(49,70)(53,71)(61,72);(9,11,15,25)(10,16,20,39)(12,22,26,50)(13,30,33,61)(14,35,31,24)(17,37,40,63)(18,44,45,53)(19,46,34,38)(21,47,49,29)(23,43,36,32)(27,57,54,72)(28,60,55,52)(41,67,65,71)(42,68,59,64)(48,69,70,56)(51,66,62,58);(9,10,15,20)(11,21,25,49)(12,17,26,40)(13,29,33,47)(14,34,31,19)(16,36,39,23)(18,43,45,32)(22,48,50,70)(24,53,35,44)(27,56,54,69)(28,59,55,42)(30,46,61,38)(37,62,63,51)(41,66,65,58)(52,71,60,67)(57,68,72,64)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 204311);

C5×#82028 #14478613