C₂×#20261574 #27156903

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂×C₆₄ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₂×#20261574

Splittings

derived, Sylow

C₉₇ ⋊ C₂×C₆₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₁₆	#87604
commutator, hypocenter	C₉₇	C₂×C₆₄
Frattini	C₁₆	C₂×#87604
Fitting	C₂×C₁₆×C₉₇	C₄
socle, prime radical	C₂²×C₉₇	C₃₂
cosocle	#15690716	C₂²
LCS	G ▷ C₉₇
UCS	C₁ ◁ C₂×C₁₆

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 163
transitive degree	μ_t(G) = 12416

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 19677);

C2×#20261574 #27156903