C₂×#20280807 #27550736

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#20280807

Splittings

Sylow

C₅₃ ⋊ C₂×#2655

Subgroups

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂³×C₅₃	C₂⁵
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₂×C₅₃×#400	C₂
socle, prime radical	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂×#400	D₅₃
hypocenter	C₅₃	C₂×#2655
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×#400

1766 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 5 generators on 79 points:

(1,2)(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)(64,66,67,70)(65,69,68,71)(74,79)(77,78);(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)(58,62)(60,63)(65,68)(69,71)(72,74,75,79)(73,77,76,78);(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)(56,58)(57,60)(59,62)(61,63)(72,73,75,76)(74,78,79,77);(56,57,59,61)(58,60,62,63)(64,65)(66,69)(67,68)(70,71)(73,76)(77,78);(54,55)

Conjugacy classes & character table

1208 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 336212);

C2×#20280807 #27550736