#34460816

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₂³×C₂²⋊C₄×C₇₉ ⋊ C₂
Sylow	C₇₉ ⋊ #34981

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₇₉
commutator, socle, cosocle	C₂³×C₇₉	C₂⁵
Fitting	C₂³×C₂²⋊C₄×C₇₉	C₂
hypercenter	C₂³×C₂²⋊C₄	D₇₉
hypocenter	C₇₉	#34981
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₂²⋊C₄

5126 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 95
transitive degree	μ_t(G) = 2528

A minimal faithful permutation representation — 5 generators on 95 points:

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

Polycyclic presentation:

2^8,79|b^2=f,b^a=bf,c^a=cg,e^a=eh,i^a=i^78,c^b=ch,d^b=dh

Conjugacy classes & character table

3184 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 342115);