#34197261

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1312 ⋊ C₂
Sylow	C₇₉ ⋊ #11402

Subgroups

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₇₉
commutator, socle	C₂²×C₇₉	C₂×C₄×C₈
Fitting	C₇₉×#1312	C₂
prime radical	C₂²×D₇₉	C₄×C₈
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	#1312	D₇₉
hypocenter	C₇₉	#11402
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄×C₈ ◁ #1312

322 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 163 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:

(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)(89,93)(94,99)(95,100)(101,103);(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)(81,84)(85,87)(88,89,90,94,91,95,96,101,92,93,97,99,98,100,102,103);(80,81,82,85,83,84,86,87)

Polycyclic presentation:

2^8,79|a^2=d,b^2=e,d^2=f,e^2=g,g^2=h,b^a=bf,c^b=ch,i^c=i^78

Conjugacy classes & character table

3232 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 78560);