#27154839

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₉₇ ⋊ #3407

Subgroups

	subgroup	G/H
center, Frattini	C₈	C₂³×D₉₇
commutator, socle	C₂×C₉₇	C₂³×C₈
Fitting	C₂×#131×C₉₇	C₂
prime radical	C₂×#41×C₉₇	C₄
cosocle	C₈×C₉₇	C₂⁴
hypercenter	C₂×#131	D₉₇
hypocenter	C₉₇	#3407
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₈ ◁ C₂×#131

360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 129
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 4 generators on 129 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)(99,105)(104,114)(106,117)(107,118)(115,124)(116,125)(119,127)(126,129);(100,108)(104,114)(109,120)(110,121)(115,124)(116,125)(122,128)(126,129);(98,100)(99,104)(101,109)(102,110)(103,108)(105,114)(106,115)(107,116)(111,122)(112,120)(113,121)(117,124)(118,125)(119,126)(123,128)(127,129);(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)(98,99,101,106,102,107,111,119,103,105,112,117,113,118,123,127)(100,104,109,115,110,116,122,126,108,114,120,124,121,125,128,129)

Polycyclic presentation:

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

Conjugacy classes & character table

1988 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 17613);