#34198088

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

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #11610

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#2960	C₂
socle	C₂²×C₇₉	C₂×C₂²⋊C₈
prime radical	C₂³×C₇₉	C₂²⋊C₈
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	#2960	D₇₉
hypocenter	C₇₉	#11610
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2960

326 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

2260 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 79387);