C₂×#20272925 #27305128

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃₂ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂×#20272925

Splittings

Fitting	C₂×C₅₃×C₆₄ ⋊ C₂
Sylow	C₅₃ ⋊ C₂×M₁₂₈

Subgroups

	subgroup	G/H
center	C₂×C₃₂	C₂×D₅₃
commutator	C₂×C₅₃	C₂²×C₃₂
Frattini	C₃₂	C₂²×D₅₃
Fitting	C₂×C₅₃×C₆₄	C₂
socle	C₂²×C₅₃	C₂×C₃₂
prime radical	C₂²×D₅₃	C₃₂
cosocle	C₃₂×C₅₃	C₂³
hypercenter	C₂×C₆₄	D₅₃
hypocenter	C₅₃	C₂×M₁₂₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₃₂ ◁ C₂×C₆₄

134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 119 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)(57,63)(64,78)(65,79)(66,80)(67,81)(82,98)(83,99)(84,100)(85,101)(86,102)(87,103)(104,113)(105,114)(106,115)(107,116)(117,119);(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)(56,57,58,64,59,65,68,82,60,66,69,83,72,85,88,104,61,67,70,84,73,86,89,105,75,87,91,106,94,107,108,117,62,63,71,78,74,79,90,98,76,80,92,99,95,101,109,113,77,81,93,100,96,102,110,114,97,103,111,115,112,116,118,119);(54,55)

Conjugacy classes & character table

3488 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 90604);

C2×#20272925 #27305128