#27140035

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

Aliases

No aliases for this group.

Splittings

Fitting	C₈²×C₉₇ ⋊ C₂
Sylow	C₉₇ ⋊ #1454

Subgroups

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₉₇
commutator, socle	C₂²×C₉₇	C₂×C₄²
Fitting	C₈²×C₉₇	C₂
prime radical	C₂²×D₉₇	C₄²
cosocle	C₄²×C₉₇	C₂³
hypercenter	C₈²	D₉₇
hypocenter	C₉₇	#1454
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄² ◁ C₈²

178 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 129
transitive degree	μ_t(G) = 6208

A minimal faithful permutation representation — 3 generators on 129 points:

(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)(99,104)(100,106)(105,111)(107,112)(115,120)(119,125)(121,127)(126,129);(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)(98,100)(99,103)(101,107)(102,106)(104,110)(105,109)(108,112)(111,113)(114,116,117,122,118,123,124,128)(115,119,121,126,120,125,127,129);(98,99,101,105,102,104,108,111)(100,103,107,109,106,110,112,113)(114,115)(116,119)(117,121)(118,120)(122,126)(123,125)(124,127)(128,129)

Polycyclic presentation:

2^7,97|a^2=d,b^2=e,d^2=f,e^2=g,c^a=cfg,c^b=cf,h^c=h^96

Conjugacy classes & character table

3128 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 2809);