#34176079

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

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×M₁₂₈ ⋊ C₂
Sylow	C₇₉ ⋊ #7513

Subgroups

	subgroup	G/H
center	C₁₆	(C₂×D₇₉)⋊C₄
commutator	C₂²×C₇₉	C₂×C₃₂
Frattini	C₂×C₃₂	C₂×D₇₉
Fitting	C₇₉×M₁₂₈	C₂
socle	C₂×C₇₉	C₂²⋊C₃₂
prime radical	C₂²×D₇₉	C₃₂
cosocle	C₂×C₃₂×C₇₉	C₂²
hypercenter	M₁₂₈	D₇₉
hypocenter	C₇₉	#7513
LCS	G ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₁₆ ◁ C₂×C₃₂ ◁ M₁₂₈

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

3208 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 57378);