C₂×#20275606 #27388037

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₂×#20275606

Splittings

Fitting	C₂²×#108×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1975

Subgroups

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂²×#108×C₅₃	C₂
socle	C₂³×C₅₃	C₄×D₄
prime radical	C₂²×#13133239	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂²×#108	D₅₃
hypocenter	C₅₃	C₂×#1975
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

1518 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 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)(60,62,64,73,65,71,63,72)(61,67,66,70,69,75,68,74);(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)(57,59)(61,66,69,68)(62,71)(67,74,75,70)(72,73);(56,57,58,59)(60,61,64,68,65,69,63,66)(62,70,73,67,71,74,72,75);(54,55)

Conjugacy classes & character table

1520 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 173513);

C2×#20275606 #27388037