C₄×#15690900 #27158818

order	12416
automorphism	?
homology	Abelianization (H₁) = C₄×C₃₂ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₄×#15690900

Splittings

derived, Sylow

C₉₇ ⋊ C₄×C₃₂

Subgroups

	subgroup	G/H
center, hypercenter	C₄²	#11715906
commutator, hypocenter	C₉₇	C₄×C₃₂
Frattini	C₂×C₄	C₂×#11715906
Fitting	C₄²×C₉₇	C₈
socle, prime radical	C₂²×C₉₇	C₂×C₁₆
cosocle	C₂×#13095514	C₂²
LCS	G ▷ C₉₇
UCS	C₁ ◁ C₄²

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 133
transitive degree	μ_t(G) = 12416

A minimal faithful permutation representation — 2 generators on 133 points:

(2,3,4,7,5,8,10,16)(6,12,13,15,14,22,28,30)(9,19,20,40,21,41,43,32)(11,24,25,48,26,49,23,46)(17,33,34,59,35,18,37,38)(27,51,52,65,44,71,53,81)(29,45,54,72,36,61,47,75)(31,55,56,82,57,39,63,64)(42,67,68,80,69,91,70,74)(50,77,78,89,79,66,83,62)(58,84,85,60,86,73,76,87)(88,95,94,90,92,96,93,97)(102,103,104,108,105,109,112,118,106,110,113,119,115,121,124,128,107,111,114,120,116,122,125,129,117,123,126,130,127,131,132,133);(1,2,6,15,31,58,82,93,78,63,69,41,66,85,46,74,71,24,50,72,54,25,4,11,27,43,70,89,59,84,61,12,21,8,18,39,65,90,96,67,92,87,37,52,13,29,35,16,32,40,7,17,36,28,53,34,60,88,91,95,97,81,55,33,3,9,22,45,73,38,62,68,20,44,26,10,23,47,75,79,49,51,80,48,76,77,19,42,56,83,94,64,86,57,30,14,5)(98,99,100,101)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 21592);

C4×#15690900 #27158818