C₄×#41×C₆₇ #16863045

order	4288
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₆₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₄×#41×C₆₇

Subgroups

	subgroup	G/H
center	C₄²×C₆₇	C₂²
commutator	C₂	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂⁴
prime radical	C₂×#41×C₆₇	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₆₇ ◁ G

310 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 2144

A minimal faithful permutation representation — 4 generators on 79 points:

(1,2,3,4)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79);(5,6)(7,10)(8,9)(11,12);(6,9)(10,12);(5,7,8,11)(6,10,9,12)

Conjugacy classes & character table

2680 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₆₇×#1476, C₆₇×#1477, C₆₇×#1488, …
maximal normal subgroup of	C₆₇×#1476, C₆₇×#1477, C₆₇×#1488, …
maximal quotient of	C₄×#117×C₆₇, C₄²×D₄×C₆₇, C₄²×Q₈×C₆₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4288, 198);

C4×#41×C67 #16863045