#34208494

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

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1402 ⋊ C₂
Sylow	C₇₉ ⋊ #13562

Subgroups

	subgroup	G/H
center	C₂	AΣL(1,𝔽₁₆)×D₇₉
commutator	C₂×D₄×C₇₉	C₂²×C₄
Frattini	#127	C₂²×D₇₉
Fitting	C₇₉×#1402	C₂
socle	C₂×C₇₉	C₂×AΣL(1,𝔽₁₆)
prime radical	#23772745	C₂
cosocle	#127×C₇₉	C₂³
hypercenter	#1402	D₇₉
hypocenter	C₇₉	#13562
LCS	G ▷ C₂×D₄×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #127 ◁ #1402

490 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 95
transitive degree	μ_t(G) = 1264

A minimal faithful permutation representation — 3 generators on 95 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

691 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 89793);