#34175183

order	20224
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #7171

Subgroups

	subgroup	G/H
center	C₂²×C₄	#11929609
commutator	C₈×C₇₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₉
Fitting	C₂×#335×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	#102
cosocle	C₂×C₄×C₈×C₇₉	C₂²
hypercenter	C₂×#335	D₇₉
hypocenter	C₇₉	#7171
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

282 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 107
transitive degree	μ_t(G) = 20224

A minimal faithful permutation representation — 2 generators on 107 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)(84,86,88,91,89,90,87,85)(92,93,95,99,96,100,94,98,97,101,102,106,103,105,104,107);(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)(84,85,87,90,89,91,88,86)(93,98)(94,102)(95,104)(96,103)(99,100)(101,107)(105,106)

Polycyclic presentation:

2^8,79|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=fg,f^2=h,b^a=bc,c^a=cf,e^a=efh,f^a=fh,i^b=i^78

Conjugacy classes & character table

2272 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 56482);