C₄×#16810342 #27271895

order	13568
automorphism	?
homology	Abelianization (H₁) = C₄×C₆₄ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₄×#16810342

Splittings

derived, Sylow

C₅₃ ⋊ C₄×C₆₄

Subgroups

	subgroup	G/H
center, hypercenter	C₄×C₃₂	D₅₃
commutator, hypocenter	C₅₃	C₄×C₆₄
Frattini	C₂×C₃₂	C₂×D₅₃
Fitting	C₄×C₃₂×C₅₃	C₂
socle, prime radical	C₂²×C₅₃	C₂×C₃₂
cosocle	C₂×C₃₂×C₅₃	C₂²
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₄×C₃₂

86 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 121
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 2 generators on 121 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)(58,59,60,65,61,66,70,80,62,67,71,81,74,84,90,100,63,68,72,82,75,85,91,101,77,87,93,103,96,106,110,115,64,69,73,83,76,86,92,102,78,88,94,104,97,107,111,116,79,89,95,105,98,108,112,117,99,109,113,118,114,119,120,121);(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)(54,55,56,57)

Conjugacy classes & character table

3584 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 57371);

C4×#16810342 #27271895