C₂×C₇₉⋊C₁₂₈ #34176178

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₂₈ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₇₉⋊C₁₂₈
unambiguous semidirect product	(C₂×C₇₉)⋊C₁₂₈

Splittings

derived, Sylow

C₇₉ ⋊ C₂×C₁₂₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₆₄	D₇₉
commutator, hypocenter	C₇₉	C₂×C₁₂₈
Frattini	C₆₄	C₂×D₇₉
Fitting	C₂×C₆₄×C₇₉	C₂
socle, prime radical	C₂²×C₇₉	C₆₄
cosocle	C₆₄×C₇₉	C₂²
LCS	G ▷ C₇₉
UCS	C₁ ◁ C₂×C₆₄

46 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 209
transitive degree	μ_t(G) = 20224

A minimal faithful permutation representation — 2 generators on 209 points:

(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)(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)(82,83,84,90,85,91,96,111,86,92,97,112,101,116,126,146,87,93,98,113,102,117,127,147,105,120,130,150,136,156,166,181,88,94,99,114,103,118,128,148,106,121,131,151,137,157,167,182,108,123,133,153,139,159,169,184,142,162,172,187,176,191,196,202,89,95,100,115,104,119,129,149,107,122,132,152,138,158,168,183,109,124,134,154,140,160,170,185,143,163,173,188,177,192,197,203,110,125,135,155,141,161,171,186,144,164,174,189,178,193,198,204,145,165,175,190,179,194,199,205,180,195,200,206,201,207,208,209);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(80,81)

Conjugacy classes & character table

5248 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 57477);

C2×C79⋊C128 #34176178