#27156917

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₂²⋊C₄×C₉₇ ⋊ C₄
Sylow	C₉₇ ⋊ #1437

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂²×#87604
commutator	C₂²×C₉₇	C₂×C₄²
Fitting	C₂×C₂²⋊C₄×C₉₇	C₄
socle	C₂³×C₉₇	C₂²×C₄
prime radical	C₂⁴×D₉₇	C₂²
cosocle	C₂³×D₉₇	C₂³
hypercenter	C₂×C₂²⋊C₄	#87604
hypocenter	C₉₇	#1437
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

828 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 828 conjugacy classes of subgroups — too many to draw. Only its 164 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) = 109
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 3 generators on 109 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)(107,109);(98,99,100,101)(103,105);(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)(102,103)(104,105)(106,107,108,109)

Polycyclic presentation:

2^7,97|a^2=f,b^2=g,c^2=ef,b^a=bd,c^a=ce,h^a=h^75,h^c=h^22,h^f=h^96

Conjugacy classes & character table

536 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 19691);