#34205143

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂×#350×C₇₉ ⋊ C₂
Sylow	C₇₉ ⋊ #12980

Subgroups

	subgroup	G/H
center	C₂×C₈	D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#350×C₇₉	C₂
socle	C₂²×C₇₉	C₈×D₄
prime radical	C₂×#14579181	C₄
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#350	D₇₉
hypocenter	C₇₉	#12980
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#350

470 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)(84,87)(85,93)(86,100)(88,98)(89,108)(90,104)(91,92)(94,111)(95,106)(96,103)(97,107)(99,115)(101,114)(102,112)(105,110)(109,113);(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)(81,83)(84,86,99,103,88,106,110,93,91,102,109,111,90,101,89,107)(85,92,112,113,94,104,114,108,97,87,100,115,96,98,95,105);(80,81,82,83)(84,85,89,95,90,96,109,100,91,97,110,114,88,94,99,112)(86,98,93,113,101,87,103,105,102,104,107,115,106,92,111,108)

Polycyclic presentation:

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

Conjugacy classes & character table

2260 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 86442);