D₃₂×D₉₇ #27144032

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

direct product

D₃₂×D₉₇

Splittings

Fitting	D₃₂×C₉₇ ⋊ C₂
Sylow	C₉₇ ⋊ C₂×D₃₂

Subgroups

	subgroup	G/H
center	C₂	D₁₆×D₉₇
commutator, cosocle	C₁₆×C₉₇	C₂³
Frattini	C₁₆	C₂²×D₉₇
Fitting	D₃₂×C₉₇	C₂
socle	C₂×C₉₇	C₂×D₁₆
hypercenter	D₃₂	D₉₇
hypocenter	C₉₇	C₂×D₃₂
LCS	G ▷ C₁₆×C₉₇ ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ D₃₂

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 129
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 3 generators on 129 points:

(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);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(99,104)(100,109)(101,110)(102,114)(105,106)(107,123)(108,119)(111,128)(112,126)(113,117)(115,127)(116,118)(120,129)(121,122)(124,125);(98,99)(100,104)(101,113)(102,116)(103,108)(105,109)(106,110)(107,114)(111,123)(112,119)(115,129)(117,125)(118,124)(120,128)(121,126)(122,127)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 6806);

D32×D97 #27144032