C₂×#20265619 #27184262

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#20265619

Splittings

Fitting	C₂×Q₃₂×C₁₀₁ ⋊ C₂
Sylow	C₁₀₁ ⋊ C₂×#510

Subgroups

	subgroup	G/H
center	C₂²	D₈×D₁₀₁
commutator, cosocle	C₈×C₁₀₁	C₂⁴
Frattini	C₈	C₂³×D₁₀₁
Fitting	C₂×Q₃₂×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂×D₈
prime radical	C₂×#15692278	C₂
hypercenter	C₂×Q₃₂	D₁₀₁
hypocenter	C₁₀₁	C₂×#510
LCS	G ▷ C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

360 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 135
transitive degree	μ_t(G) = 6464

A minimal faithful permutation representation — 4 generators on 135 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)(105,111)(110,125)(112,127)(113,123)(114,131)(118,130)(126,129)(128,134);(104,106,109,117)(105,110,111,125)(107,119,122,135)(108,124,120,116)(112,128,127,134)(113,129,123,126)(114,130,131,118)(115,132,133,121);(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)(104,105)(106,114)(107,118)(108,123)(109,111)(110,115)(112,121)(113,120)(116,134)(117,131)(119,129)(122,130)(124,128)(125,133)(126,135)(127,132);(102,103)

Conjugacy classes & character table

1132 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 17640);