C₂×D₃₅₈₉ #28385171

order	14356
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₇₁₇₈
direct product	C₂×D₃₅₈₉
unambiguous semidirect product	(C₂×C₃₇)⋊D₉₇
unambiguous semidirect product	(C₂×C₉₇)⋊D₃₇

Splittings

Fitting, socle	C₂×C₃₇×C₉₇ ⋊ C₂
derived, Hall	C₃₇×C₉₇ ⋊ C₂²
Sylow	C₉₇ ⋊ C₂×D₃₇
Sylow	C₃₇ ⋊ C₂×D₉₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃₅₈₉
commutator, cosocle, hypocenter	C₃₇×C₉₇	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₃₇×C₉₇	C₂
LCS	G ▷ C₃₇×C₉₇
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 136
transitive degree	μ_t(G) = 7178

A minimal faithful permutation representation — 2 generators on 136 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)(40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136);(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(41,136)(42,135)(43,134)(44,133)(45,132)(46,131)(47,130)(48,129)(49,128)(50,127)(51,126)(52,125)(53,124)(54,123)(55,122)(56,121)(57,120)(58,119)(59,118)(60,117)(61,116)(62,115)(63,114)(64,113)(65,112)(66,111)(67,110)(68,109)(69,108)(70,107)(71,106)(72,105)(73,104)(74,103)(75,102)(76,101)(77,100)(78,99)(79,98)(80,97)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,90)(88,89)

Conjugacy classes & character table

3592 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₄₃₅₆
maximal normal subgroup of	D₁₄₃₅₆
maximal quotient of	D₁₄₃₅₆, Dic₇₁₇₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(14356, 13);

C2×D3589 #28385171