C₂×D₆₁₁₁ #37567807

order	24444
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₂₂₂₂
direct product	C₂×D₆₁₁₁
unambiguous semidirect product	(C₂×C₇)⋊D₈₇₃
unambiguous semidirect product	(C₂×C₉)⋊D₆₇₉
unambiguous semidirect product	(C₂×C₉₇)⋊D₆₃

Splittings

Fitting	C₂×C₇×C₉×C₉₇ ⋊ C₂
derived, Hall	C₇×C₉×C₉₇ ⋊ C₂²
Hall	C₉×C₉₇ ⋊ C₂×D₇
Hall	C₇×C₉₇ ⋊ C₂×D₉
Sylow	C₉₇ ⋊ C₂×D₆₃
Hall	C₇×C₉ ⋊ C₂×D₉₇
Sylow	C₉ ⋊ C₂×D₆₇₉
Sylow	C₇ ⋊ C₂×D₈₇₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₆₁₁₁
commutator, cosocle, hypocenter	C₇×C₉×C₉₇	C₂²
Frattini	C₃	C₂×D₂₀₃₇
Fitting	C₂×C₇×C₉×C₉₇	C₂
socle	C₂×C₃×C₇×C₉₇	D₃
LCS	G ▷ C₇×C₉×C₉₇
UCS	C₁ ◁ C₂

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 115
transitive degree	μ_t(G) = 12222

A minimal faithful permutation representation — 2 generators on 115 points:

(1,2)(3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(4,11)(5,10)(6,9)(7,8)(13,18)(14,17)(15,16)(20,115)(21,114)(22,113)(23,112)(24,111)(25,110)(26,109)(27,108)(28,107)(29,106)(30,105)(31,104)(32,103)(33,102)(34,101)(35,100)(36,99)(37,98)(38,97)(39,96)(40,95)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)

Conjugacy classes & character table

6114 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₄₄₄₄
maximal normal subgroup of	D₂₄₄₄₄
maximal quotient of	Dic₁₂₂₂₂, D₂₄₄₄₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24444, 70);

C2×D6111 #37567807