#34200040

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #12049

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	D₄×C₁₆×C₇₉	C₂
socle	C₂²×C₇₉	C₈×D₄
prime radical	C₂×D₄×C₇₉	C₂×C₈
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	D₄×C₁₆	D₇₉
hypocenter	C₇₉	#12049
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ D₄×C₁₆

226 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 91 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:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(88,90,92,97,93,98,100,103,91,89,94,95,99,96,101,102);(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)(80,81,83,85)(82,87,86,84)(89,90)(95,97)(96,98)(102,103);(81,84)(82,86)(85,87)(88,89,92,95,93,96,100,102,91,90,94,97,99,98,101,103)

Polycyclic presentation:

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

Conjugacy classes & character table

3196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 81339);