#27155845

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₄×#41×C₉₇ ⋊ C₂
Sylow	C₉₇ ⋊ #3524

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₉₇
commutator, socle, cosocle	C₂²×C₉₇	C₂⁵
Fitting	C₄×#41×C₉₇	C₂
hypercenter	C₄×#41	D₉₇
hypocenter	C₉₇	#3524
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄×#41

1030 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 121
transitive degree	μ_t(G) = 6208

A minimal faithful permutation representation — 5 generators on 121 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)(98,100,101,104)(99,103,102,105)(106,110,111,109)(107,113,114,119)(108,117,118,116)(112,120,115,121);(106,109,111,110)(107,113,114,119)(108,117,118,116)(112,121,115,120);(99,102)(103,105)(106,109,111,110)(107,113,114,119)(108,116,118,117)(112,120,115,121);(106,108,111,118)(107,112,114,115)(109,116,110,117)(113,120,119,121);(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)(100,103)(101,102)(104,105)(106,107)(108,115)(109,113)(110,119)(111,114)(112,118)(116,121)(117,120)

Polycyclic presentation:

2^7,97|b^2=f,c^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,e^a=ef,h^a=h^96,d^b=df

Conjugacy classes & character table

1964 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 18619);