C₂×#20261603 #27157407

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂×#20261603

Splittings

Fitting	C₂²×C₈×C₉₇ ⋊ C₄
Sylow	C₉₇ ⋊ C₂×H(3,ℤ/4)

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×#87604
commutator	C₄×C₉₇	C₂×C₄²
Frattini	C₂×C₄	C₂²×#87604
Fitting	C₂²×C₈×C₉₇	C₄
socle	C₂²×C₉₇	#95
prime radical	C₂³×D₉₇	D₄
cosocle	C₂×C₄×D₉₇	C₂³
hypercenter	C₂²×C₈	#87604
hypocenter	C₉₇	C₂×H(3,ℤ/4)
LCS	G ▷ C₄×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈

404 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 115
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 3 generators on 115 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 20181);