C₂×#23761528 #34203259

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#23761528

Splittings

Sylow

C₇₉ ⋊ C₂×#1395

Subgroups

	subgroup	G/H
center	C₂×C₄	#14579075
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂×#41	C₂²×D₇₉
Fitting	C₂×#410×C₇₉	C₂
socle	C₂²×C₇₉	#342
prime radical	C₂²×#41×C₇₉	C₂²
cosocle	C₂×#41×C₇₉	C₂³
hypercenter	C₂×#410	D₇₉
hypocenter	C₇₉	C₂×#1395
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ C₂×#410

766 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 113
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 3 generators on 113 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)(82,84,87,97,88,98,85,95)(83,89,92,105,93,106,90,94)(86,100,101,103,102,96,99,111)(91,104,108,112,109,113,107,110);(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)(82,83,86,91)(84,94,96,110)(85,92,99,108)(87,90,101,107)(88,93,102,109)(89,103,104,95)(97,106,111,113)(98,105,100,112);(80,81)

Conjugacy classes & character table

1768 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 84558);

C2×#23761528 #34203259