C₂×D₇₄₉ #15688482

order	2996
automorphism	Aut(G) order 952728
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₄₉₈
direct product	C₂×D₇₄₉
unambiguous semidirect product	(C₂×C₇)⋊D₁₀₇
unambiguous semidirect product	(C₂×C₁₀₇)⋊D₇

Splittings

Fitting, socle	C₂×C₇×C₁₀₇ ⋊ C₂
derived, Hall	C₇×C₁₀₇ ⋊ C₂²
Sylow	C₁₀₇ ⋊ C₂×D₇
Sylow	C₇ ⋊ C₂×D₁₀₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₇₄₉
commutator, cosocle, hypocenter	C₇×C₁₀₇	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₁₀₇	C₂
LCS	G ▷ C₇×C₁₀₇
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 116
transitive degree	μ_t(G) = 1498

A minimal faithful permutation representation — 2 generators on 116 points:

(1,2)(3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116);(4,9)(5,8)(6,7)(11,116)(12,115)(13,114)(14,113)(15,112)(16,111)(17,110)(18,109)(19,108)(20,107)(21,106)(22,105)(23,104)(24,103)(25,102)(26,101)(27,100)(28,99)(29,98)(30,97)(31,96)(32,95)(33,94)(34,93)(35,92)(36,91)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₅×D₇₄₉, C₂×D₃₇₄₅, C₂×C₅×D₇₄₉, …
maximal normal subgroup of	C₂×C₅×D₇₄₉, C₂×C₇×D₇₄₉, C₂×C₁₁×D₇₄₉, …
maximal quotient of	D₅×D₇₄₉, C₂×D₃₇₄₅, C₂×C₅×D₇₄₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2996, 7);

C2×D749 #15688482