#34203238

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #12581

Subgroups

	subgroup	G/H
center	C₂×C₄	#14579075
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂²×D₄	C₂²×D₇₉
Fitting	C₂×#410×C₇₉	C₂
socle	C₂²×C₇₉	C₂×#99
prime radical	C₂²×#41×C₇₉	C₂²
cosocle	C₂²×D₄×C₇₉	C₂³
hypercenter	C₂×#410	D₇₉
hypocenter	C₇₉	#12581
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ C₂×#410

854 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 99
transitive degree	μ_t(G) = 2528

A minimal faithful permutation representation — 3 generators on 99 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)(84,86,88,94)(85,90,92,98)(87,93,95,99)(89,96,91,97);(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)(85,89,92,91)(90,96,98,97);(84,85)(86,90)(87,91,95,89)(88,92)(93,97,99,96)(94,98)

Polycyclic presentation:

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

Conjugacy classes & character table

1768 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 84537);