C₂×#23766245 #34447479

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂×#23766245

Splittings

Sylow

C₇₉ ⋊ C₂×#2342

Subgroups

	subgroup	G/H
center	C₂⁴	C₂³×D₇₉
commutator	C₂²×C₇₉	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₇₉
Fitting	C₂³×C₂²⋊C₄×C₇₉	C₂
socle	C₂⁴×C₇₉	C₂⁴
prime radical	C₂⁶×C₇₉	C₂²
cosocle	C₂³×C₇₉	C₂⁵
hypercenter	C₂³×C₂²⋊C₄	D₇₉
hypocenter	C₇₉	C₂×#2342
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₂²⋊C₄

4990 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 97
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 5 generators on 97 points:

(8,10)(19,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21);(12,16)(15,18);(7,8)(9,10)(11,13,14,17)(12,15,16,18)(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)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97);(3,4,5,6)(8,10)(11,12)(13,15)(14,16)(17,18);(1,2)

Conjugacy classes & character table

3220 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 328778);

C2×#23766245 #34447479