#34204199

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#2150 ⋊ C₂
Sylow	C₇₉ ⋊ #12786

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄
commutator	C₂³×C₇₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#2150	C₂
socle	C₂²×C₇₉	C₂×C₂²⋊C₈
prime radical	C₂³×D₇₉	C₂×C₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#2150	D₇₉
hypocenter	C₇₉	#12786
LCS	G ▷ C₂³×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2150

454 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 115
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 3 generators on 115 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)(85,91)(86,94)(87,97)(93,104)(96,107)(98,109)(101,110)(112,115);(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)(80,81,82,83)(84,86)(85,90)(87,95)(88,96)(89,94)(91,100)(92,101)(93,102)(97,106)(98,108)(99,107)(103,110)(104,111)(105,112)(109,114)(113,115);(81,83)(84,85,87,92,88,93,98,105,89,91,97,103,99,104,109,113)(86,90,95,101,96,102,108,112,94,100,106,110,107,111,114,115)

Polycyclic presentation:

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

Conjugacy classes & character table

2248 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 85498);