#34205685

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

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #13069

Subgroups

	subgroup	G/H
center	C₂	#23763829
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#473×C₇₉	C₂
socle	C₂×C₇₉	#1893
prime radical	C₂×#41×C₇₉	D₄
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#473	D₇₉
hypocenter	C₇₉	#13069
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#473

450 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)(84,104)(85,108)(86,101)(87,102)(88,111)(90,118)(91,117)(92,116)(93,120)(94,125)(96,131)(97,130)(98,129)(99,109)(100,106)(103,135)(105,136)(107,128)(110,137)(112,134)(113,140)(114,141)(115,119)(121,142)(122,126)(123,139)(124,138)(127,133)(132,143);(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)(80,82,94,124,86,97,84,96,87,98,105,132,85,107,106,133)(81,88,110,121,91,113,90,93,92,114,103,122,109,112,119,123)(83,95,125,138,101,130,104,131,102,129,136,143,108,128,100,127)(89,111,137,142,117,140,118,120,116,141,135,126,99,134,115,139);(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)(83,99,102,117)(84,103)(85,91)(86,109)(87,92)(88,96)(89,101,116,108)(90,105)(94,119)(95,126,129,142)(97,121)(98,123)(100,115,104,118)(106,110)(107,122)(111,138,141,143)(112,132)(113,124)(114,133)(120,128,139,130)(125,135,136,137)(127,134,131,140)

Polycyclic presentation:

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

Conjugacy classes & character table

1294 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 86984);