C₄²×Q₈×C₁₃₇ #31777283

order	17536
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄²×C₁₃₇ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₄²×Q₈×C₁₃₇

Subgroups

	subgroup	G/H
center	C₂×C₄²×C₁₃₇	C₂²
commutator	C₂	C₂²×C₄²×C₁₃₇
Frattini, cosocle	C₂³	C₂⁴×C₁₃₇
socle, prime radical	C₂³×C₁₃₇	C₂⁴
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₁₃₇ ◁ G

684 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 153
transitive degree	μ_t(G) = 17536

A minimal faithful permutation representation — 4 generators on 153 points:

(1,2,3,4)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153);(5,6,7,8);(9,10,12,14)(11,15,16,13);(9,11,12,16)(10,13,14,15)

Conjugacy classes & character table

10960 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₄×Q₈⋊C₈×C₁₃₇, C₁₃₇×#8171, C₁₃₇×#8173, …
maximal normal subgroup of	C₄×Q₈⋊C₈×C₁₃₇, C₁₃₇×#8171, C₁₃₇×#8173, …
maximal quotient of	C₄²×C₄⋊C₄×C₁₃₇, C₄×#387×C₁₃₇, C₄×#389×C₁₃₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17536, 1004);

C42×Q8×C137 #31777283