#34203344

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×#347×C₇₉ ⋊ C₂
Sylow	C₇₉ ⋊ #12609

Subgroups

	subgroup	G/H
center	C₄	C₄×(C₂×D₇₉)⋊C₄
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#347×C₇₉	C₂
socle	C₂×C₇₉	#1749
prime radical	C₂×#11929624	C₂×C₄
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#347	D₇₉
hypocenter	C₇₉	#12609
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂×#97 ◁ C₂×#347

444 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 2528

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

Polycyclic presentation:

2^8,79|a^2=e,b^2=f,c^2=e,f^2=g,g^2=h,b^a=bd,d^a=dh,i^a=i^78,c^b=cg,e^b=eh,f^c=fh

Conjugacy classes & character table

1774 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 84643);