#27144347

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	#385×C₉₇ ⋊ C₂
Sylow	C₉₇ ⋊ #2315

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₉₇
commutator	C₂²×C₉₇	C₂³×C₄
Fitting	#385×C₉₇	C₂
socle, cosocle	C₂³×C₉₇	C₂⁴
prime radical	C₂⁴×D₉₇	C₂²
hypercenter	#385	D₉₇
hypocenter	C₉₇	#2315
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ #385

816 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 113
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 4 generators on 113 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)(107,111)(110,112);(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)(102,103,104,105)(108,113)(110,112);(106,108)(107,110)(109,113)(111,112);(98,99,100,101)(103,105)(106,107)(108,112)(109,111)(110,113)

Polycyclic presentation:

2^7,97|a^2=g,c^2=f,b^a=be,c^a=cf,d^a=de,c^b=ce,h^d=h^96

Conjugacy classes & character table

1388 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 7121);