#34200079

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

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1370 ⋊ C₂
Sylow	C₇₉ ⋊ SD₁₆⋊C₁₆

Subgroups

	subgroup	G/H
center	C₂×C₈	D₄×D₇₉
commutator	C₄×C₇₉	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#1370	C₂
socle	C₂²×C₇₉	C₈×D₄
prime radical	C₂×D₃₁₆	C₂×C₈
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	#1370	D₇₉
hypocenter	C₇₉	SD₁₆⋊C₁₆
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1370

232 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 93 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) = 10112

A minimal faithful permutation representation — 3 generators on 103 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)(81,85)(84,86)(88,90,91,97,92,98,99,103,93,89,94,95,100,96,101,102);(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)(81,84)(82,87)(85,86);(80,81)(82,84)(83,85)(86,87)(88,89,91,95,92,96,99,102,93,90,94,97,100,98,101,103)

Polycyclic presentation:

2^8,79|a^2=e,c^2=e,d^2=f,e^2=g,g^2=h,b^a=bd,c^a=cf,d^a=df,d^b=df,i^b=i^78,i^c=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, 81378);