Finite Group Databasebeta

← Prev (#11764498) Next (#11764500) →

C₅×#6402 #11764499

order	1120
isoclinism	isoclinic class of #6400
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₁

Aliases

Splittings

Fitting	C₅×C₇×Q₁₆ ⋊ C₂
Hall	C₅×C₇ ⋊ SD₃₂
Sylow	C₇ ⋊ C₅×SD₃₂

Subgroups

	subgroup	G/H
center	C₂×C₅	#1141
commutator, cosocle	C₇×C₈	C₂²×C₅
Frattini	C₈	C₂×C₅×D₇
Fitting	C₅×C₇×Q₁₆	C₂
socle	C₂×C₅×C₇	D₈
prime radical	C₅×D₅₆	C₂
hypercenter	C₅×Q₁₆	D₇
hypocenter	C₇	C₅×SD₃₂
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×C₈ ◁ C₅×Q₁₆

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 560

A minimal faithful permutation representation — 2 generators on 28 points:

(1,2,3,4,5)(6,7,9,11,12,10,8)(13,14,17,20)(15,22,24,28)(16,25,23,19)(18,26,27,21);(7,8)(9,10)(11,12)(14,18)(15,21)(16,23)(19,28)(20,27)(22,25)(24,26)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1120, 105);