#34174925

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1312 ⋊ C₂
Sylow	C₇₉ ⋊ #7070

Subgroups

	subgroup	G/H
center	C₂×C₄	#14579083
commutator	C₈×C₇₉	C₂×C₁₆
Frattini	C₈²	C₂×D₇₉
Fitting	C₇₉×#1312	C₂
socle	C₂²×C₇₉	D₄⋊C₈
prime radical	C₂×D₆₃₂	C₈
cosocle	C₈²×C₇₉	C₂²
hypercenter	#1312	D₇₉
hypocenter	C₇₉	#7070
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1312

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

normal subgroups only

This group has 156 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) = 111
transitive degree	μ_t(G) = 10112

A minimal faithful permutation representation — 2 generators on 111 points:

(1,2)(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)(97,101)(98,107)(99,110)(102,103)(104,111)(105,106)(108,109);(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)(80,81,82,85,83,86,88,91,84,87,89,92,90,93,94,95)(96,97,99,103,100,105,110,104)(98,106,108,102,109,101,107,111)

Polycyclic presentation:

2^8,79|a^2=d,c^2=e,d^2=f,e^2=g,f^2=h,b^a=bc,c^a=ce,e^a=eg,i^a=i^78,c^b=ceg,d^b=dg,e^b=eg,i^b=i^78

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, 56224);