#34201047

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

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #12213

Subgroups

	subgroup	G/H
center	C₂²	#19098749
commutator, cosocle	C₂×C₄⋊C₄×C₇₉	C₂³
Frattini	C₂×C₄⋊C₄	C₂²×D₇₉
Fitting	C₇₉×#1611	C₂
socle	C₂²×C₇₉	B(4,𝔽₂)
prime radical	C₂³×C₇₉	#120
hypercenter	#1611	D₇₉
hypocenter	C₇₉	#12213
LCS	G ▷ C₂×C₄⋊C₄×C₇₉ ▷ C₂³×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #1611

380 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 103
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 3 generators on 103 points:

(9,12,15,21)(10,17,20,18)(11,13,14,23)(16,24,22,19)(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)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97)(98,99)(100,101)(102,103);(2,5)(3,8)(6,7)(9,11,15,14)(10,16,20,22)(12,23,21,13)(17,19,18,24)(25,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,103,101,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27);(1,2,4,7)(3,5,8,6)(9,10,15,20)(11,19,14,24)(12,22,21,16)(13,17,23,18)

Polycyclic presentation:

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

Conjugacy classes & character table

925 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 82346);