C₂×#20261842 #27158965

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃₂ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#20261842

Splittings

derived, Sylow

C₉₇ ⋊ C₂²×C₃₂

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₈	#11715906
commutator, hypocenter	C₉₇	C₂²×C₃₂
Frattini	C₄	C₂²×#11715906
Fitting	C₂×C₈×C₉₇	C₈
socle	C₂²×C₉₇	C₂×C₁₆
prime radical	C₂²×D₉₇	C₁₆
cosocle	#13095514	C₂³
LCS	G ▷ C₉₇
UCS	C₁ ◁ C₂×C₈

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 131
transitive degree	μ_t(G) = 6208

A minimal faithful permutation representation — 3 generators on 131 points:

(1,2)(3,17)(4,6)(5,23)(7,9)(8,32)(10,11)(12,45)(13,15)(14,53)(16,18)(19,66)(20,22)(21,70)(24,51)(25,27)(26,47)(28,29)(30,31)(33,64)(34,36)(35,84)(37,39)(38,52)(40,42)(43,44)(46,76)(48,50)(49,74)(54,75)(55,63)(56,58)(57,68)(59,60)(61,83)(62,93)(65,73)(67,97)(69,87)(71,80)(72,78)(77,86)(79,94)(81,88)(82,85)(89,90)(91,92)(95,96);(2,3,5,8,4,7,10,16)(6,12,14,15,13,22,28,30)(9,19,21,41,20,40,43,32)(11,24,26,49,25,48,23,46)(17,33,35,18,34,38,59,37)(27,51,52,63,44,71,53,81)(29,45,54,72,39,65,47,75)(31,55,57,82,56,36,61,62)(42,67,69,80,68,91,70,74)(50,77,79,87,78,66,83,60)(58,84,86,64,85,73,76,89)(88,92,96,93,95,90,97,94)(100,101,102,106,103,107,110,116,104,108,111,117,113,119,122,126,105,109,112,118,114,120,123,127,115,121,124,128,125,129,130,131);(98,99)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 21739);

C2×#20261842 #27158965