#27150001

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

Aliases

No aliases for this group.

Splittings

Fitting	#399×C₉₇ ⋊ C₂
Sylow	C₉₇ ⋊ #2842

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₉₇
commutator, socle, cosocle	C₂³×C₉₇	C₂⁴
Fitting	#399×C₉₇	C₂
prime radical	C₂⁴×D₉₇	C₂²
hypercenter	#399	D₉₇
hypocenter	C₉₇	#2842
LCS	G ▷ C₂³×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ #399

548 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 125 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) = 121
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 4 generators on 121 points:

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

Polycyclic presentation:

2^7,97|b^2=eg,c^2=g,b^a=be,c^a=cf,h^a=h^96,c^b=cg,d^b=def,d^c=de

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, 12775);