C₂×#20266016 #27186299

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂×C₆₄ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₂×#20266016

Splittings

derived, Sylow

C₁₀₁ ⋊ C₂×C₆₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₁₆	#87938
commutator, hypocenter	C₁₀₁	C₂×C₆₄
Frattini	C₁₆	C₂×#87938
Fitting	C₂×C₁₆×C₁₀₁	C₄
socle, prime radical	C₂²×C₁₀₁	C₃₂
cosocle	#15692161	C₂²
LCS	G ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₁₆

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 167
transitive degree	μ_t(G) = 12928

A minimal faithful permutation representation — 2 generators on 167 points:

(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,28)(14,21,22,24)(16,25,26,20)(19,30,31,45)(23,34,35,41)(27,38,39,37)(29,42,43,33)(32,47,48,65)(36,51,52,62)(40,55,56,58)(44,59,60,54)(46,49,63,50)(53,66,67,80)(57,70,71,77)(61,74,75,73)(64,68,78,69)(72,81,82,91)(76,85,86,88)(79,83,89,84)(87,92,93,98)(90,94,96,95)(97,99,100,101)(104,105,106,111,107,112,116,126,108,113,117,127,120,130,136,146,109,114,118,128,121,131,137,147,123,133,139,149,142,152,156,161,110,115,119,129,122,132,138,148,124,134,140,150,143,153,157,162,125,135,141,151,144,154,158,163,145,155,159,164,160,165,166,167);(1,2,5,10,19,32,49,42,25,12,3,7,14,23,36,53,68,59,38,21,8,16,27,40,57,72,83,74,55,34,17,29,44,61,76,87,94,85,70,51,30,46,64,79,90,97,99,92,81,66,47,65,80,91,98,101,100,96,89,78,63,45,62,77,88,95,93,86,75,60,43,28,41,58,73,84,82,71,56,39,26,15,24,37,54,69,67,52,35,22,13,6,11,20,33,50,48,31,18,9,4)(102,103)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 19677);

C2×#20266016 #27186299