#34199478

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₂×#364×C₇₉ ⋊ C₂
Sylow	C₇₉ ⋊ #11920

Subgroups

	subgroup	G/H
center	C₄	#19098306
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₂×#364×C₇₉	C₂
socle	C₂×C₇₉	C₂×#331
prime radical	#23773828	C₂
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	C₂×#364	D₇₉
hypocenter	C₇₉	#11920
LCS	G ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 2528

A minimal faithful permutation representation — 3 generators on 111 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(80,83,85,97)(81,87,90,89)(82,92,95,94)(84,96,99,107)(86,100,102,91)(88,103,106,105)(93,98,110,109)(101,104,111,108);(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)(80,82)(81,86)(83,92)(84,98)(85,95)(87,100)(88,104)(89,91)(90,102)(93,107)(94,97)(96,110)(99,109)(101,105)(103,111)(106,108);(80,81,84,88,83,87,96,103,85,90,99,106,97,89,107,105)(82,91,93,108,94,102,109,111,95,100,110,104,92,86,98,101)

Polycyclic presentation:

2^8,79|a^2=e,c^2=h,d^2=g,e^2=fg,f^2=h,b^a=bd,d^a=df,f^a=fh,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,i^b=i^78

Conjugacy classes & character table

2230 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 80777);