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G = C23.589C24order 128 = 27

306th central stem extension by C23 of C24

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C23.589C24, C24.396C23, C22.3632+ 1+4, C22.2692- 1+4, C22⋊C4.13D4, C23.211(C2×D4), C2.60(D46D4), C2.94(D45D4), (C23×C4).454C22, (C22×C4).871C23, (C2×C42).643C22, C22.398(C22×D4), C23.8Q8.45C2, C23.11D4.33C2, C23.34D4.26C2, (C22×Q8).181C22, C23.83C2376C2, C23.67C2378C2, C23.81C2380C2, C23.78C2342C2, C24.C22.48C2, C23.65C23118C2, C23.63C23129C2, C2.C42.296C22, C2.57(C23.38C23), C2.11(C22.57C24), C2.60(C22.33C24), C2.54(C22.50C24), C2.38(C22.35C24), (C2×C4).94(C2×D4), (C2×C22⋊Q8).42C2, (C2×C4).191(C4○D4), (C2×C4⋊C4).403C22, C22.451(C2×C4○D4), (C2×C422C2).12C2, (C2×C22⋊C4).256C22, SmallGroup(128,1421)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.589C24
C1C2C22C23C24C23×C4C23.34D4 — C23.589C24
C1C23 — C23.589C24
C1C23 — C23.589C24
C1C23 — C23.589C24

Generators and relations for C23.589C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=f2=a, e2=ba=ab, ac=ca, ede-1=ad=da, geg=ae=ea, af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, gdg=abd, fg=gf >

Subgroups: 420 in 228 conjugacy classes, 96 normal (82 characteristic)
C1, C2 [×7], C2 [×2], C4 [×18], C22 [×7], C22 [×10], C2×C4 [×8], C2×C4 [×42], Q8 [×4], C23, C23 [×2], C23 [×6], C42 [×3], C22⋊C4 [×4], C22⋊C4 [×9], C4⋊C4 [×20], C22×C4 [×14], C22×C4 [×5], C2×Q8 [×5], C24, C2.C42 [×14], C2×C42 [×2], C2×C22⋊C4 [×6], C2×C4⋊C4 [×11], C22⋊Q8 [×4], C422C2 [×4], C23×C4, C22×Q8, C23.34D4, C23.8Q8, C23.63C23, C24.C22, C23.65C23 [×2], C23.67C23, C23.78C23, C23.11D4 [×2], C23.81C23, C23.83C23 [×2], C2×C22⋊Q8, C2×C422C2, C23.589C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], 2+ 1+4, 2- 1+4 [×3], C23.38C23, C22.33C24, C22.35C24, D45D4, D46D4, C22.50C24, C22.57C24, C23.589C24

Smallest permutation representation of C23.589C24
On 64 points
Generators in S64
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)
(1 9)(2 10)(3 11)(4 12)(5 39)(6 40)(7 37)(8 38)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)(33 62)(34 63)(35 64)(36 61)
(1 53)(2 54)(3 55)(4 56)(5 52)(6 49)(7 50)(8 51)(9 27)(10 28)(11 25)(12 26)(13 31)(14 32)(15 29)(16 30)(17 36)(18 33)(19 34)(20 35)(21 40)(22 37)(23 38)(24 39)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)
(1 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)
(1 19 11 45)(2 18 12 48)(3 17 9 47)(4 20 10 46)(5 58 37 30)(6 57 38 29)(7 60 39 32)(8 59 40 31)(13 51 43 21)(14 50 44 24)(15 49 41 23)(16 52 42 22)(25 61 53 34)(26 64 54 33)(27 63 55 36)(28 62 56 35)
(1 43 3 41)(2 16 4 14)(5 20 7 18)(6 45 8 47)(9 15 11 13)(10 44 12 42)(17 38 19 40)(21 36 23 34)(22 62 24 64)(25 31 27 29)(26 58 28 60)(30 56 32 54)(33 52 35 50)(37 46 39 48)(49 61 51 63)(53 59 55 57)
(2 12)(4 10)(5 39)(6 8)(7 37)(14 44)(16 42)(17 19)(18 46)(20 48)(21 23)(22 50)(24 52)(26 54)(28 56)(30 58)(32 60)(33 62)(34 36)(35 64)(38 40)(45 47)(49 51)(61 63)

G:=sub<Sym(64)| (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64), (1,9)(2,10)(3,11)(4,12)(5,39)(6,40)(7,37)(8,38)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,55)(26,56)(27,53)(28,54)(29,59)(30,60)(31,57)(32,58)(33,62)(34,63)(35,64)(36,61), (1,53)(2,54)(3,55)(4,56)(5,52)(6,49)(7,50)(8,51)(9,27)(10,28)(11,25)(12,26)(13,31)(14,32)(15,29)(16,30)(17,36)(18,33)(19,34)(20,35)(21,40)(22,37)(23,38)(24,39)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (1,2,3,4)(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), (1,19,11,45)(2,18,12,48)(3,17,9,47)(4,20,10,46)(5,58,37,30)(6,57,38,29)(7,60,39,32)(8,59,40,31)(13,51,43,21)(14,50,44,24)(15,49,41,23)(16,52,42,22)(25,61,53,34)(26,64,54,33)(27,63,55,36)(28,62,56,35), (1,43,3,41)(2,16,4,14)(5,20,7,18)(6,45,8,47)(9,15,11,13)(10,44,12,42)(17,38,19,40)(21,36,23,34)(22,62,24,64)(25,31,27,29)(26,58,28,60)(30,56,32,54)(33,52,35,50)(37,46,39,48)(49,61,51,63)(53,59,55,57), (2,12)(4,10)(5,39)(6,8)(7,37)(14,44)(16,42)(17,19)(18,46)(20,48)(21,23)(22,50)(24,52)(26,54)(28,56)(30,58)(32,60)(33,62)(34,36)(35,64)(38,40)(45,47)(49,51)(61,63)>;

G:=Group( (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64), (1,9)(2,10)(3,11)(4,12)(5,39)(6,40)(7,37)(8,38)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,55)(26,56)(27,53)(28,54)(29,59)(30,60)(31,57)(32,58)(33,62)(34,63)(35,64)(36,61), (1,53)(2,54)(3,55)(4,56)(5,52)(6,49)(7,50)(8,51)(9,27)(10,28)(11,25)(12,26)(13,31)(14,32)(15,29)(16,30)(17,36)(18,33)(19,34)(20,35)(21,40)(22,37)(23,38)(24,39)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (1,2,3,4)(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), (1,19,11,45)(2,18,12,48)(3,17,9,47)(4,20,10,46)(5,58,37,30)(6,57,38,29)(7,60,39,32)(8,59,40,31)(13,51,43,21)(14,50,44,24)(15,49,41,23)(16,52,42,22)(25,61,53,34)(26,64,54,33)(27,63,55,36)(28,62,56,35), (1,43,3,41)(2,16,4,14)(5,20,7,18)(6,45,8,47)(9,15,11,13)(10,44,12,42)(17,38,19,40)(21,36,23,34)(22,62,24,64)(25,31,27,29)(26,58,28,60)(30,56,32,54)(33,52,35,50)(37,46,39,48)(49,61,51,63)(53,59,55,57), (2,12)(4,10)(5,39)(6,8)(7,37)(14,44)(16,42)(17,19)(18,46)(20,48)(21,23)(22,50)(24,52)(26,54)(28,56)(30,58)(32,60)(33,62)(34,36)(35,64)(38,40)(45,47)(49,51)(61,63) );

G=PermutationGroup([(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32),(33,35),(34,36),(37,39),(38,40),(41,43),(42,44),(45,47),(46,48),(49,51),(50,52),(53,55),(54,56),(57,59),(58,60),(61,63),(62,64)], [(1,9),(2,10),(3,11),(4,12),(5,39),(6,40),(7,37),(8,38),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,55),(26,56),(27,53),(28,54),(29,59),(30,60),(31,57),(32,58),(33,62),(34,63),(35,64),(36,61)], [(1,53),(2,54),(3,55),(4,56),(5,52),(6,49),(7,50),(8,51),(9,27),(10,28),(11,25),(12,26),(13,31),(14,32),(15,29),(16,30),(17,36),(18,33),(19,34),(20,35),(21,40),(22,37),(23,38),(24,39),(41,57),(42,58),(43,59),(44,60),(45,61),(46,62),(47,63),(48,64)], [(1,2,3,4),(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)], [(1,19,11,45),(2,18,12,48),(3,17,9,47),(4,20,10,46),(5,58,37,30),(6,57,38,29),(7,60,39,32),(8,59,40,31),(13,51,43,21),(14,50,44,24),(15,49,41,23),(16,52,42,22),(25,61,53,34),(26,64,54,33),(27,63,55,36),(28,62,56,35)], [(1,43,3,41),(2,16,4,14),(5,20,7,18),(6,45,8,47),(9,15,11,13),(10,44,12,42),(17,38,19,40),(21,36,23,34),(22,62,24,64),(25,31,27,29),(26,58,28,60),(30,56,32,54),(33,52,35,50),(37,46,39,48),(49,61,51,63),(53,59,55,57)], [(2,12),(4,10),(5,39),(6,8),(7,37),(14,44),(16,42),(17,19),(18,46),(20,48),(21,23),(22,50),(24,52),(26,54),(28,56),(30,58),(32,60),(33,62),(34,36),(35,64),(38,40),(45,47),(49,51),(61,63)])

32 conjugacy classes

class 1 2A···2G2H2I4A···4P4Q···4V
order12···2224···44···4
size11···1444···48···8

32 irreducible representations

dim11111111111112244
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2D4C4○D42+ 1+42- 1+4
kernelC23.589C24C23.34D4C23.8Q8C23.63C23C24.C22C23.65C23C23.67C23C23.78C23C23.11D4C23.81C23C23.83C23C2×C22⋊Q8C2×C422C2C22⋊C4C2×C4C22C22
# reps11111211212114813

Matrix representation of C23.589C24 in GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
420000
010000
004000
000400
000011
000034
,
200000
020000
000100
001000
000033
000002
,
100000
140000
001000
000400
000030
000003
,
100000
140000
001000
000100
000010
000034

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,2,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,3,0,0,0,0,1,4],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,3,2],[1,1,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[1,1,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,3,0,0,0,0,0,4] >;

C23.589C24 in GAP, Magma, Sage, TeX

C_2^3._{589}C_2^4
% in TeX

G:=Group("C2^3.589C2^4");
// GroupNames label

G:=SmallGroup(128,1421);
// by ID

G=gap.SmallGroup(128,1421);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,344,758,723,100,1571,346]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=f^2=a,e^2=b*a=a*b,a*c=c*a,e*d*e^-1=a*d=d*a,g*e*g=a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=c*e=e*c,c*f=f*c,c*g=g*c,g*d*g=a*b*d,f*g=g*f>;
// generators/relations

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