#34176126

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1408 ⋊ C₂
Sylow	C₇₉ ⋊ #7529

Subgroups

	subgroup	G/H
center	C₂	#23761562
commutator	C₄×C₈×C₇₉	C₂×C₄
Frattini	#493	C₂×D₇₉
Fitting	C₇₉×#1408	C₂
socle	C₂×C₇₉	#1408
prime radical	#23763014	C₂
cosocle	#493×C₇₉	C₂²
hypercenter	#1408	D₇₉
hypocenter	C₇₉	#7529
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #493 ◁ #1408

186 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 25 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 — 2 generators on 95 points:

(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)(81,86)(82,90)(83,84)(87,88)(89,95)(91,92)(93,94);(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)(80,81)(82,87,90,86)(83,88,84,93)(85,89)(91,95,92,94)

Polycyclic presentation:

2^8,79|a^2=d,c^2=fg,e^2=gh,f^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,c^b=cfgh,d^b=def,e^b=egh,f^b=fh,i^b=i^78,d^c=dfg,e^d=egh,f^d=fh

Conjugacy classes & character table

688 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 57425);