C₂×D₄×#11719949 #28291056

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×D₄×#11719949

Splittings

Fitting	C₂×D₄×Q₈×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₂×D₄×#41

Subgroups

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂×D₄×Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁵
hypercenter	C₂×D₄×Q₈	D₅₃
hypocenter	C₅₃	C₂×D₄×#41
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×D₄×Q₈

12314 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 6 generators on 67 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)(60,62,63,66)(61,64,65,67);(60,61,63,65)(62,67,66,64);(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)(62,66)(64,67);(57,59);(56,57)(58,59);(54,55)

Conjugacy classes & character table

1400 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 1076532);

C2×D4×#11719949 #28291056