#27153512

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	#444×C₉₇ ⋊ C₂
Sylow	C₉₇ ⋊ #3289

Subgroups

	subgroup	G/H
center	C₂	D₄×D₃₈₈
commutator, cosocle	C₂×C₄×C₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	#444×C₉₇	C₂
socle	C₂×C₉₇	D₄²
hypercenter	#444	D₉₇
hypocenter	C₉₇	#3289
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #444

448 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)(98,101)(99,107)(100,113)(102,119)(103,116)(104,117)(105,121)(106,108)(109,123)(110,125)(111,127)(112,118)(114,129)(115,128)(120,122)(124,126);(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)(98,101)(99,106)(100,112)(102,116)(103,119)(104,117)(105,120)(107,108)(109,125)(110,123)(111,127)(113,118)(114,128)(115,129)(121,122)(124,126);(98,100,104,115)(99,105,110,122)(101,112,117,129)(102,118,103,114)(106,120,123,121)(107,124,125,127)(108,126,109,111)(113,119,128,116);(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)(98,99,104,110)(100,111,115,126)(101,107,117,125)(102,109,103,108)(105,118,122,114)(106,119,123,116)(112,120,129,121)(113,127,128,124)

Polycyclic presentation:

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

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, 16286);