C₂×#23761495 #34203161

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

Aliases

direct product

C₂×#23761495

Splittings

Sylow

C₇₉ ⋊ C₂×#1382

Subgroups

	subgroup	G/H
center	C₂²×C₄	#11929609
commutator	C₈×C₇₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#427×C₇₉	C₂
socle	C₂³×C₇₉	#102
prime radical	C₂⁴×C₇₉	SD₁₆
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#427	D₇₉
hypocenter	C₇₉	C₂×#1382
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

590 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 117
transitive degree	μ_t(G) = 10112

A minimal faithful permutation representation — 3 generators on 117 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)(86,88,90,99,103,114,104,115,92,101,89,98,91,100,106,116)(87,93,94,97,107,111,105,117,96,109,110,113,95,108,102,112);(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)(82,83,84,85)(86,87)(88,97)(89,102)(90,105)(91,107)(92,96)(93,98)(94,104)(95,103)(99,109)(100,117)(101,113)(106,110)(108,116)(111,115)(112,114);(80,81)

Conjugacy classes & character table

2272 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 84460);

C2×#23761495 #34203161