#34205141

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #12979

Subgroups

	subgroup	G/H
center	C₈	#14579129
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#2955	C₂
socle	C₂×C₇₉	#1869
prime radical	#575×C₇₉	C₄
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#2955	D₇₉
hypocenter	C₇₉	#12979
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2955

456 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 143
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 3 generators on 143 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)(80,83)(81,89)(82,94)(84,103)(85,105)(86,100)(87,99)(88,108)(90,116)(91,118)(92,98)(93,113)(95,125)(96,127)(97,122)(101,107)(102,123)(104,131)(106,112)(109,129)(110,138)(111,135)(114,120)(115,121)(117,130)(119,133)(124,128)(126,140)(132,134)(136,139)(137,142)(141,143);(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)(80,82,85,96,86,95,106,126,87,97,107,128,84,102,104,132)(81,88,91,110,92,109,119,137,93,111,120,139,90,115,117,141)(83,94,105,127,100,125,112,140,99,122,101,124,103,123,131,134)(89,108,118,138,98,129,133,142,113,135,114,136,116,121,130,143);(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)(80,81,85,91,86,92,106,119,87,93,107,120,84,90,104,117)(82,88,96,110,95,109,126,137,97,111,128,139,102,115,132,141)(83,98,101,130,100,113,131,118,99,116,105,133,103,89,112,114)(94,121,124,142,125,108,134,136,122,129,127,143,123,135,140,138)

Polycyclic presentation:

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

Conjugacy classes & character table

1780 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 86440);