#34176074

order	20224
automorphism	?
homology	Abelianization (H₁) = C₄×C₃₂ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #7510

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₃₂	C₂×D₇₉
commutator	C₂×C₇₉	C₄×C₃₂
Fitting	C₄×C₃₂×C₇₉	C₂
socle, prime radical	C₂²×C₇₉	C₂×C₃₂
cosocle	C₂×C₃₂×C₇₉	C₂²
hypercenter	C₄×C₃₂	D₇₉
hypocenter	C₇₉	#7510
LCS	G ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₃₂ ◁ C₄×C₃₂

82 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 147
transitive degree	μ_t(G) = 20224

A minimal faithful permutation representation — 2 generators on 147 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)(80,81,82,83)(85,91)(92,106)(93,107)(94,108)(95,109)(110,126)(111,127)(112,128)(113,129)(114,130)(115,131)(132,141)(133,142)(134,143)(135,144)(145,147);(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)(84,85,86,92,87,93,96,110,88,94,97,111,100,113,116,132,89,95,98,112,101,114,117,133,103,115,119,134,122,135,136,145,90,91,99,106,102,107,118,126,104,108,120,127,123,129,137,141,105,109,121,128,124,130,138,142,125,131,139,143,140,144,146,147)

Polycyclic presentation:

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

Conjugacy classes & character table

5152 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 57373);