C₇×C₅⋊D₅×D₃₇ #39906017

order	25900
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₂×C₅

Aliases

direct product	C₇×C₅⋊D₅×D₃₇
unambiguous semidirect product	(C₅×D₃₇)⋊(C₇×D₅)

Splittings

Fitting, socle, Hall	C₅²×C₇×C₃₇ ⋊ C₂²
derived, Hall	C₅²×C₃₇ ⋊ C₂²×C₇
Hall	C₇×C₃₇ ⋊ C₂×C₅⋊D₅
Hall	C₅²×C₇ ⋊ C₂×D₃₇
Sylow	C₃₇ ⋊ C₂×C₇×C₅⋊D₅
Sylow	C₅² ⋊ C₂×C₇×D₃₇

Subgroups

	subgroup	G/H
center, hypercenter	C₇	C₅⋊D₅×D₃₇
commutator, cosocle, hypocenter	C₅²×C₃₇	C₂²×C₇
Frattini	C₁	G
Fitting, socle	C₅²×C₇×C₃₇	C₂²
LCS	G ▷ C₅²×C₃₇
UCS	C₁ ◁ C₇

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 54
transitive degree	μ_t(G) = 6475

A minimal faithful permutation representation — 3 generators on 54 points:

(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)(43,44,46,47,45)(48,49,50,51,52,53,54);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(39,40)(41,42)(44,45)(46,47);(38,39,41,42,40)

Conjugacy classes & character table

1960 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25900, 87);

C7×C5⋊D5×D37 #39906017