C₇×#19127078 #49150733

order	37100
automorphism	?
homology	Abelianization (H₁) = C₄×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₇×#19127078

Splittings

Fitting, socle, Hall	C₅²×C₇×C₅₃ ⋊ C₄
derived, Hall	C₅²×C₅₃ ⋊ C₄×C₇
Hall	C₇×C₅₃ ⋊ #1043
Hall	C₅²×C₇ ⋊ #6157
Sylow	C₅₃ ⋊ C₇×#1043
Sylow	C₅² ⋊ C₇×#6157

Subgroups

	subgroup	G/H
center, hypercenter	C₇	#19127078
commutator, hypocenter	C₅²×C₅₃	C₄×C₇
Frattini	C₁	G
Fitting, socle	C₅²×C₇×C₅₃	C₄
prime radical	C₇×C₅⋊D₂₆₅	C₂
cosocle	C₅⋊D₂₆₅	C₂×C₇
LCS	G ▷ C₅²×C₅₃
UCS	C₁ ◁ C₇

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 70
transitive degree	μ_t(G) = 9275

A minimal faithful permutation representation — 3 generators on 70 points:

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

Conjugacy classes & character table

2345 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37100, 95);

C7×#19127078 #49150733