C₂×C₃₇⋊(D₄₁⋊C₅) #44476830

order	30340
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₃₇⋊(D₄₁⋊C₅)
unambiguous semidirect product	(C₂×C₃₇)⋊(D₄₁⋊C₅)
unambiguous semidirect product	(C₂×C₄₁⋊C₅)⋊D₃₇
unambiguous semidirect product	(C₂×D₁₅₁₇)⋊C₅

Splittings

Fitting, socle	C₂×C₃₇×C₄₁ ⋊ C₂×C₅
derived, Hall	C₃₇×C₄₁ ⋊ C₂²×C₅
Hall	C₃₇×C₄₁⋊C₅ ⋊ C₂²
Hall	C₂×D₁₅₁₇ ⋊ C₅
Hall	C₄₁⋊C₅ ⋊ C₂×D₃₇
Sylow	C₄₁ ⋊ C₂×C₅×D₃₇
Sylow	C₃₇ ⋊ C₂×D₄₁⋊C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	C₃₇⋊(D₄₁⋊C₅)
commutator, cosocle, hypocenter	C₃₇×C₄₁	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₃₇×C₄₁	C₂×C₅
LCS	G ▷ C₃₇×C₄₁
UCS	C₁ ◁ C₂

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 3034

A minimal faithful permutation representation — 2 generators on 80 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30340, 22);

C2×C37⋊(D41⋊C5) #44476830