#34203147

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1380 ⋊ C₂
Sylow	C₇₉ ⋊ #12556

Subgroups

	subgroup	G/H
center	C₂³	D₈×D₇₉
commutator	C₈×C₇₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1380	C₂
socle	C₂³×C₇₉	C₂×D₈
prime radical	C₂³×D₇₉	D₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1380	D₇₉
hypocenter	C₇₉	#12556
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 10112

A minimal faithful permutation representation — 3 generators on 119 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)(88,91,94,107)(89,96,100,99)(90,102,105,118)(92,93,106,108)(95,109,112,111)(97,113,114,98)(101,115,116,110)(103,104,117,119);(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)(84,85,86,87)(88,90,93,104,91,102,106,117,94,105,108,119,107,118,92,103)(89,95,98,110,99,111,114,116,100,112,113,115,96,109,97,101);(80,81,82,83)(88,89)(90,101)(91,96)(92,98)(93,97)(94,100)(95,103)(99,107)(102,115)(104,109)(105,116)(106,113)(108,114)(110,118)(111,119)(112,117)

Polycyclic presentation:

2^8,79|a^2=e,b^2=f,c^2=h,d^2=gh,f^2=g,g^2=h,b^a=bd,d^a=dg,f^a=fgh,g^a=gh,i^c=i^78

Conjugacy classes & character table

1804 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 84446);