#34205431

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₄×#135×C₇₉ ⋊ C₂
Sylow	C₇₉ ⋊ #13027

Subgroups

	subgroup	G/H
center	C₄	#19098758
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₄×#135×C₇₉	C₂
socle	C₂×C₇₉	#1879
prime radical	#23776534	C₂
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₄×#135	D₇₉
hypocenter	C₇₉	#13027
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₄×#135

484 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)(80,82)(81,89)(83,101)(84,95)(85,96)(86,104)(87,99)(88,105)(90,94)(91,106)(92,108)(93,107)(97,103)(98,102)(100,111)(109,110);(80,82)(81,88,93,108)(83,101)(84,95)(85,96)(86,104)(87,99)(89,105,107,92)(90,106,100,109)(91,111,110,94)(97,103)(98,102);(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)(80,81,85,90,86,92,103,110,87,93,83,100,102,105,84,91)(82,94,97,107,98,106,96,108,99,111,95,89,104,109,101,88)

Polycyclic presentation:

2^8,79|a^2=f,b^2=dfg,d^2=g,e^2=gh,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,i^a=i^78,d^b=dg,e^b=eg,f^b=fg,g^b=gh,d^c=dg,e^c=eg,f^c=fg,g^c=gh

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, 86730);