C₂×#20260523 #27143369

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#20260523

Splittings

Fitting	C₂×#100×C₉₇ ⋊ C₂
Sylow	C₉₇ ⋊ C₂×AΣL(1,𝔽₁₆)

Subgroups

	subgroup	G/H
center	C₂²	#15690724
commutator	C₂³×C₉₇	C₂²×C₄
Frattini	C₂×D₄	C₂²×D₉₇
Fitting	C₂×#100×C₉₇	C₂
socle	C₂²×C₉₇	#99
prime radical	C₂×#15690778	C₂
cosocle	C₂×D₄×C₉₇	C₂³
hypercenter	C₂×#100	D₉₇
hypocenter	C₉₇	C₂×AΣL(1,𝔽₁₆)
LCS	G ▷ C₂³×C₉₇ ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ C₂×#100

438 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 69 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) = 107
transitive degree	μ_t(G) = 1552

A minimal faithful permutation representation — 3 generators on 107 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)(101,104)(105,106);(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)(100,101)(102,106,107,104)(103,105);(98,99)

Conjugacy classes & character table

1082 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 6143);