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G = C23.728C24  order 128 = 27

445th central stem extension by C23 of C24

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

Aliases: C24.110C23, C23.728C24, C22.5012+ (1+4), C23⋊2D4⋊51C2, (C22×C4).239C23, (C2×C42).736C22, C23.10D4⋊114C2, C23.11D4⋊134C2, C24.3C22⋊97C2, (C22×D4).303C22, C24.C22⋊178C2, C2.18(C24⋊C22), C23.83C23⋊136C2, C2.117(C22.32C24), C2.51(C22.54C24), C2.C42.431C22, C2.68(C22.34C24), (C2×C4).252(C4○D4), (C2×C4⋊C4).537C22, C22.576(C2×C4○D4), (C2×C22⋊C4).346C22, SmallGroup(128,1560)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C23 — C23.728C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C22×D4 — C23⋊2D4 — C23.728C24
Lower central
C1 — C23 — C23.728C24
Upper central
C1 — C23 — C23.728C24
Jennings
C1 — C23 — C23.728C24

Subgroups: 644 in 254 conjugacy classes, 84 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×12], C22 [×3], C22 [×4], C22 [×28], C2×C4 [×2], C2×C4 [×32], D4 [×24], C23, C23 [×28], C42, C22⋊C4 [×18], C4⋊C4 [×4], C22×C4 [×3], C22×C4 [×8], C2×D4 [×19], C24 [×4], C2.C42 [×2], C2.C42 [×6], C2×C42, C2×C22⋊C4 [×16], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C22×D4 [×2], C22×D4 [×4], C24.C22 [×2], C24.3C22, C23⋊2D4 [×4], C23.10D4, C23.10D4 [×4], C23.11D4 [×2], C23.83C23, C23.728C24

Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×2], C24, C2×C4○D4, 2+ (1+4) [×6], C22.32C24, C22.34C24 [×2], C22.54C24 [×3], C24⋊C22, C23.728C24

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

Smallest permutation representation
►On 64 points
Generators in S64
(1 9)(2 10)(3 11)(4 12)(5 38)(6 39)(7 40)(8 37)(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 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 63)(34 64)(35 61)(36 62)

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

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

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

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

Matrix representation ►G ⊆ GL10(𝔽5)

1000000000
0100000000
0040000000
0004000000
0000400000
0000040000
0000004000
0000000400
0000000040
0000000004
,
1000000000
0100000000
0040000000
0004000000
0000400000
0000040000
0000001000
0000000100
0000000010
0000000001
,
4000000000
0400000000
0040000000
0004000000
0000400000
0000040000
0000004000
0000000400
0000000040
0000000004
,
3000000000
0300000000
0000400000
0000040000
0010000000
0001000000
0000003000
0000000300
0000000020
0000000002
,
0100000000
1000000000
0000100000
0000010000
0010000000
0001000000
0000000010
0000000001
0000001000
0000000100
,
1000000000
0400000000
0001000000
0010000000
0000040000
0000400000
0000000100
0000001000
0000000004
0000000040
,
1000000000
0100000000
0001000000
0040000000
0000010000
0000400000
0000000100
0000004000
0000000004
0000000010

G:=sub<GL(10,GF(5))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,2],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0] >;
 

Character table of C23.728C24

 class 12A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L4M4N
 size 11111111888844444488888888
ρ111111111111111111111111111    trivial
ρ211111111-1-11-1-1-11-11-1-11-111-111    linear of order 2
ρ311111111-11-11-1-11-11-11-11-11-1-11    linear of order 2
ρ4111111111-1-1-1111111-1-1-1-111-11    linear of order 2
ρ5111111111-111-1-11-11-1-111-1-11-1-1    linear of order 2
ρ611111111-111-111111111-1-1-1-1-1-1    linear of order 2
ρ711111111-1-1-11111111-1-111-1-11-1    linear of order 2
ρ81111111111-1-1-1-11-11-11-1-11-111-1    linear of order 2
ρ911111111111-1-11-11-1-1-1-1111-1-1-1    linear of order 2
ρ1011111111-1-1111-1-1-1-111-1-1111-1-1    linear of order 2
ρ1111111111-11-1-11-1-1-1-11-111-1111-1    linear of order 2
ρ12111111111-1-11-11-11-1-111-1-11-11-1    linear of order 2
ρ13111111111-11-11-1-1-1-111-11-1-1-111    linear of order 2
ρ1411111111-1111-11-11-1-1-1-1-1-1-1111    linear of order 2
ρ1511111111-1-1-1-1-11-11-1-11111-11-11    linear of order 2
ρ161111111111-111-1-1-1-11-11-11-1-1-11    linear of order 2
ρ172-22-22-22-200002i-22i22i2i00000000    complex lifted from C4○D4
ρ182-22-22-22-200002i-22i22i2i00000000    complex lifted from C4○D4
ρ192-22-22-22-200002i22i-22i2i00000000    complex lifted from C4○D4
ρ202-22-22-22-200002i22i-22i2i00000000    complex lifted from C4○D4
ρ214-4-44-444-4000000000000000000    orthogonal lifted from 2+ (1+4)
ρ2244-444-4-4-4000000000000000000    orthogonal lifted from 2+ (1+4)
ρ234-444-4-4-44000000000000000000    orthogonal lifted from 2+ (1+4)
ρ24444-4-44-4-4000000000000000000    orthogonal lifted from 2+ (1+4)
ρ254-4-4-444-44000000000000000000    orthogonal lifted from 2+ (1+4)
ρ2644-4-4-4-444000000000000000000    orthogonal lifted from 2+ (1+4)

In GAP, Magma, Sage, TeX 

C_2^3._{728}C_2^4
 
% in TeX
 
G:=Group("C2^3.728C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1560);
 
// by ID
 
G=gap.SmallGroup(128,1560);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,723,794,185,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=1,d^2=c,g^2=a,a*b=b*a,a*c=c*a,e*d*e=a*d=d*a,a*e=e*a,g*f*g^-1=a*f=f*a,a*g=g*a,b*c=c*b,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f=c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,g*e*g^-1=a*b*e>;
 
// generators/relations
 

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