#34204707

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

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #12907

Subgroups

	subgroup	G/H
center	C₂³	D₈×D₇₉
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#366×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#366	D₇₉
hypocenter	C₇₉	#12907
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

342 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 20224

A minimal faithful permutation representation — 3 generators on 119 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)(84,85,86,87)(89,95)(94,107)(96,109)(97,106)(98,110)(103,118)(108,111)(112,117);(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)(88,90,99,113,93,102,105,116,92,101,114,119,104,115,91,100)(89,94,103,112,97,108,109,110,95,107,118,117,106,111,96,98);(80,81,82,83)(85,87)(88,89)(90,98)(91,103)(92,95)(93,106)(94,100)(96,99)(97,104)(101,110)(102,117)(105,118)(107,116)(108,119)(109,114)(111,113)(112,115)

Polycyclic presentation:

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

Conjugacy classes & character table

1792 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 86006);