#34205699

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #13071

Subgroups

	subgroup	G/H
center	C₄	#19098797
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting, prime radical	C₇₉×#3067	C₂
socle	C₂×C₇₉	#1893
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#3067	D₇₉
hypocenter	C₇₉	#13071
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #3067

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

Polycyclic presentation:

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

Conjugacy classes & character table

1300 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 86998);