C₂×D₇₇₉₁ #44528348

order	31164
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₅₅₈₂
direct product	C₂×D₇₇₉₁
unambiguous semidirect product	(C₂×C₃)⋊D₂₅₉₇
unambiguous semidirect product	(C₂×C₄₉)⋊D₁₅₉
unambiguous semidirect product	(C₂×C₅₃)⋊D₁₄₇

Splittings

Fitting	C₂×C₃×C₄₉×C₅₃ ⋊ C₂
derived, Hall	C₃×C₄₉×C₅₃ ⋊ C₂²
Hall	C₄₉×C₅₃ ⋊ C₂×D₃
Hall	C₃×C₅₃ ⋊ C₂×D₄₉
Hall	C₃×C₄₉ ⋊ C₂×D₅₃
Sylow	C₅₃ ⋊ C₂×D₁₄₇
Sylow	C₄₉ ⋊ C₂×D₁₅₉
Sylow	C₃ ⋊ C₂×D₂₅₉₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₇₇₉₁
commutator, cosocle, hypocenter	C₃×C₄₉×C₅₃	C₂²
Frattini	C₇	C₂×D₁₁₁₃
Fitting	C₂×C₃×C₄₉×C₅₃	C₂
socle	C₂×C₃×C₇×C₅₃	D₇
LCS	G ▷ C₃×C₄₉×C₅₃
UCS	C₁ ◁ C₂

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 107
transitive degree	μ_t(G) = 15582

A minimal faithful permutation representation — 2 generators on 107 points:

(1,2)(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)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107);(4,5)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(56,107)(57,106)(58,105)(59,104)(60,103)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82)

Conjugacy classes & character table

7794 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₁₁₆₄
maximal normal subgroup of	D₃₁₁₆₄
maximal quotient of	Dic₁₅₅₈₂, D₃₁₁₆₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31164, 40);

C2×D7791 #44528348