C₂×#20280071 #27555304

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#20280071

Splittings

Sylow

C₅₃ ⋊ C₂×#2580

Subgroups

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂³×C₅₃	C₂⁵
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₂×C₅₃×#390	C₂
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂³×D₄×C₅₃	C₂²
hypercenter	C₂×#390	D₅₃
hypocenter	C₅₃	C₂×#2580
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×#390

3222 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 5 generators on 75 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(60,62,63,67)(61,65,64,66);(57,59)(68,70,71,75)(69,72,73,74);(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)(61,64)(65,66)(68,69,71,73)(70,74,75,72);(56,57)(58,59)(60,61)(62,66)(63,64)(65,67);(54,55)

Conjugacy classes & character table

1532 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 340780);

C2×#20280071 #27555304