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G = C23.693C24order 128 = 27

410th central stem extension by C23 of C24

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

Aliases: C24.92C23, C23.693C24, C22.3562- 1+4, C22.4662+ 1+4, C429C437C2, (C22×C4).603C23, (C2×C42).109C22, C23.Q8.43C2, C23.11D4.57C2, C24.C22.79C2, C23.83C23124C2, C23.63C23189C2, C23.81C23130C2, C2.C42.397C22, C2.64(C22.34C24), C2.52(C22.57C24), C2.73(C22.50C24), C2.120(C22.46C24), C2.111(C22.33C24), C2.117(C22.36C24), C2.116(C22.47C24), (C2×C4).234(C4○D4), (C2×C4⋊C4).503C22, C22.554(C2×C4○D4), (C2×C22⋊C4).77C22, SmallGroup(128,1525)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.693C24
C1C2C22C23C22×C4C2×C42C24.C22 — C23.693C24
C1C23 — C23.693C24
C1C23 — C23.693C24
C1C23 — C23.693C24

Generators and relations for C23.693C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=ca=ac, e2=f2=b, g2=ba=ab, ede-1=ad=da, geg-1=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-1=abd, fg=gf >

Subgroups: 356 in 189 conjugacy classes, 88 normal (82 characteristic)
C1, C2 [×7], C2, C4 [×17], C22 [×7], C22 [×7], C2×C4 [×6], C2×C4 [×39], C23, C23 [×7], C42 [×4], C22⋊C4 [×10], C4⋊C4 [×16], C22×C4 [×14], C24, C2.C42 [×14], C2×C42 [×3], C2×C22⋊C4 [×7], C2×C4⋊C4 [×11], C429C4, C23.63C23 [×5], C24.C22 [×3], C23.Q8 [×2], C23.11D4 [×2], C23.81C23, C23.83C23, C23.693C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×6], C24, C2×C4○D4 [×3], 2+ 1+4 [×2], 2- 1+4 [×2], C22.33C24, C22.34C24, C22.36C24, C22.46C24, C22.47C24, C22.50C24, C22.57C24, C23.693C24

Smallest permutation representation of C23.693C24
On 64 points
Generators in S64
(1 10)(2 11)(3 12)(4 9)(5 37)(6 38)(7 39)(8 40)(13 52)(14 49)(15 50)(16 51)(17 46)(18 47)(19 48)(20 45)(21 43)(22 44)(23 41)(24 42)(25 54)(26 55)(27 56)(28 53)(29 60)(30 57)(31 58)(32 59)(33 64)(34 61)(35 62)(36 63)
(1 26)(2 27)(3 28)(4 25)(5 23)(6 24)(7 21)(8 22)(9 54)(10 55)(11 56)(12 53)(13 60)(14 57)(15 58)(16 59)(17 62)(18 63)(19 64)(20 61)(29 52)(30 49)(31 50)(32 51)(33 48)(34 45)(35 46)(36 47)(37 41)(38 42)(39 43)(40 44)
(1 12)(2 9)(3 10)(4 11)(5 39)(6 40)(7 37)(8 38)(13 50)(14 51)(15 52)(16 49)(17 48)(18 45)(19 46)(20 47)(21 41)(22 42)(23 43)(24 44)(25 56)(26 53)(27 54)(28 55)(29 58)(30 59)(31 60)(32 57)(33 62)(34 63)(35 64)(36 61)
(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 50 26 31)(2 16 27 59)(3 52 28 29)(4 14 25 57)(5 45 23 34)(6 17 24 62)(7 47 21 36)(8 19 22 64)(9 49 54 30)(10 15 55 58)(11 51 56 32)(12 13 53 60)(18 43 63 39)(20 41 61 37)(33 40 48 44)(35 38 46 42)
(1 36 26 47)(2 48 27 33)(3 34 28 45)(4 46 25 35)(5 58 23 15)(6 16 24 59)(7 60 21 13)(8 14 22 57)(9 17 54 62)(10 63 55 18)(11 19 56 64)(12 61 53 20)(29 43 52 39)(30 40 49 44)(31 41 50 37)(32 38 51 42)
(1 58 55 50)(2 51 56 59)(3 60 53 52)(4 49 54 57)(5 63 41 47)(6 48 42 64)(7 61 43 45)(8 46 44 62)(9 14 25 30)(10 31 26 15)(11 16 27 32)(12 29 28 13)(17 22 35 40)(18 37 36 23)(19 24 33 38)(20 39 34 21)

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

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

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

32 conjugacy classes

class 1 2A···2G2H4A···4R4S···4W
order12···224···44···4
size11···184···48···8

32 irreducible representations

dim11111111244
type+++++++++-
imageC1C2C2C2C2C2C2C2C4○D42+ 1+42- 1+4
kernelC23.693C24C429C4C23.63C23C24.C22C23.Q8C23.11D4C23.81C23C23.83C23C2×C4C22C22
# reps115322111222

Matrix representation of C23.693C24 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
200000
030000
001300
000400
000020
000002
,
010000
100000
003000
000300
000010
000004
,
400000
040000
002000
002300
000001
000010
,
010000
400000
002000
002300
000010
000001

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

C23.693C24 in GAP, Magma, Sage, TeX

C_2^3._{693}C_2^4
% in TeX

G:=Group("C2^3.693C2^4");
// GroupNames label

G:=SmallGroup(128,1525);
// by ID

G=gap.SmallGroup(128,1525);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,232,758,723,184,1571,346,192]);
// Polycyclic

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

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