C₂×C₅×D₁₃₄₉ #41008705

order	26980
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₅×D₁₃₄₉
unambiguous semidirect product	(C₂×C₁₉)⋊(C₅×D₇₁)

Splittings

Fitting, socle	C₂×C₅×C₁₉×C₇₁ ⋊ C₂
derived, Hall	C₁₉×C₇₁ ⋊ C₂²×C₅
Hall	C₅×C₁₉×C₇₁ ⋊ C₂²
Hall	C₅×C₇₁ ⋊ C₂×D₁₉
Hall	C₅×C₁₉ ⋊ C₂×D₇₁
Sylow	C₇₁ ⋊ C₂×C₅×D₁₉
Sylow	C₁₉ ⋊ C₂×C₅×D₇₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅	D₁₃₄₉
commutator, cosocle, hypocenter	C₁₉×C₇₁	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₅×C₁₉×C₇₁	C₂
LCS	G ▷ C₁₉×C₇₁
UCS	C₁ ◁ C₂×C₅

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 97
transitive degree	μ_t(G) = 13490

A minimal faithful permutation representation — 2 generators on 97 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)(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)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84)(85,86)(87,88)(89,90)(91,92,93,94,95);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(72,73,75,77,79,81,83,85,87,89,90,88,86,84,82,80,78,76,74)(96,97)

Conjugacy classes & character table

6760 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26980, 32);

C2×C5×D1349 #41008705