C₂×C₇×D₁₃₂₅ #49150669

order	37100
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₇×D₁₃₂₅
unambiguous semidirect product	(C₂×C₂₅)⋊(C₇×D₅₃)
unambiguous semidirect product	(C₂×C₅₃)⋊(C₇×D₂₅)

Splittings

Fitting	C₂×C₇×C₂₅×C₅₃ ⋊ C₂
derived, Hall	C₂₅×C₅₃ ⋊ C₂²×C₇
Hall	C₇×C₂₅×C₅₃ ⋊ C₂²
Hall	C₇×C₅₃ ⋊ C₂×D₂₅
Hall	C₇×C₂₅ ⋊ C₂×D₅₃
Sylow	C₅₃ ⋊ C₂×C₇×D₂₅
Sylow	C₂₅ ⋊ C₂×C₇×D₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₇	D₁₃₂₅
commutator, cosocle, hypocenter	C₂₅×C₅₃	C₂²×C₇
Frattini	C₅	C₂×C₇×D₂₆₅
Fitting	C₂×C₇×C₂₅×C₅₃	C₂
socle	C₂×C₅×C₇×C₅₃	D₅
LCS	G ▷ C₂₅×C₅₃
UCS	C₁ ◁ C₂×C₇

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 18550

A minimal faithful permutation representation — 2 generators on 87 points:

(3,4,5,6,7,8,9)(11,13)(12,16)(14,19)(15,22)(17,25)(18,24)(20,21)(23,29)(26,30)(27,31)(28,33)(32,34)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)(86,87);(1,2)(10,24,34,27,13,16,30,28,19,22,25,23,21,20,29,17,15,14,33,26,12,11,31,32,18)(35,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37)

Conjugacy classes & character table

9296 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37100, 31);

C2×C7×D1325 #49150669