#27139869

order	12416
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₉₇ ⋊ #1397

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂×D₉₇
commutator	C₂×C₉₇	C₄×C₁₆
Fitting	C₂×C₃₂×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	C₂×C₁₆
cosocle	C₂×C₁₆×C₉₇	C₂²
hypercenter	C₂×C₃₂	D₉₇
hypocenter	C₉₇	#1397
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₃₂

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 133
transitive degree	μ_t(G) = 12416

A minimal faithful permutation representation — 2 generators on 133 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)(98,99,100,101)(103,108)(109,118)(110,119)(111,120)(121,128)(122,129)(123,130)(131,133);(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)(102,103,104,109,105,110,112,121,106,111,113,122,115,123,124,131,107,108,114,118,116,119,125,128,117,120,126,129,127,130,132,133)

Polycyclic presentation:

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

Conjugacy classes & character table

3152 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 2643);