C₂×D₃₃₉₇ #28331123

order	13588
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₆₇₉₄
direct product	C₂×D₃₃₉₇
unambiguous semidirect product	(C₂×C₄₃)⋊D₇₉
unambiguous semidirect product	(C₂×C₇₉)⋊D₄₃

Splittings

Fitting, socle	C₂×C₄₃×C₇₉ ⋊ C₂
derived, Hall	C₄₃×C₇₉ ⋊ C₂²
Sylow	C₇₉ ⋊ C₂×D₄₃
Sylow	C₄₃ ⋊ C₂×D₇₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃₃₉₇
commutator, cosocle, hypocenter	C₄₃×C₇₉	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₄₃×C₇₉	C₂
LCS	G ▷ C₄₃×C₇₉
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 124
transitive degree	μ_t(G) = 6794

A minimal faithful permutation representation — 2 generators on 124 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)(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124);(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(47,124)(48,123)(49,122)(50,121)(51,120)(52,119)(53,118)(54,117)(55,116)(56,115)(57,114)(58,113)(59,112)(60,111)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(81,90)(82,89)(83,88)(84,87)(85,86)

Conjugacy classes & character table

3400 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₃₅₈₈
maximal normal subgroup of	D₁₃₅₈₈
maximal quotient of	D₁₃₅₈₈, Dic₆₇₉₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13588, 7);

C2×D3397 #28331123