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G = C23.729C24order 128 = 27

446th central stem extension by C23 of C24

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

Aliases: C23.729C24, C24.111C23, C22.3832- 1+4, C22.5022+ 1+4, C232D4.38C2, C23.4Q869C2, C23.Q898C2, (C2×C42).737C22, (C22×C4).240C23, C23.11D4135C2, C23.10D4115C2, C24.3C2298C2, (C22×D4).304C22, C24.C22179C2, C23.81C23137C2, C23.63C23197C2, C2.118(C22.32C24), C2.52(C22.54C24), C2.C42.432C22, C2.56(C22.56C24), C2.69(C22.34C24), C2.124(C22.36C24), (C2×C4).253(C4○D4), (C2×C4⋊C4).538C22, C22.577(C2×C4○D4), (C2×C22⋊C4).347C22, SmallGroup(128,1561)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.729C24
C1C2C22C23C22×C4C22×D4C232D4 — C23.729C24
C1C23 — C23.729C24
C1C23 — C23.729C24
C1C23 — C23.729C24

Generators and relations for C23.729C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=1, d2=ca=ac, g2=a, ab=ba, 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 >

Subgroups: 532 in 225 conjugacy classes, 84 normal (82 characteristic)
C1, C2 [×7], C2 [×3], C4 [×13], C22 [×7], C22 [×21], C2×C4 [×2], C2×C4 [×35], D4 [×12], C23, C23 [×21], C42, C22⋊C4 [×16], C4⋊C4 [×9], C22×C4 [×12], C2×D4 [×10], C24 [×3], C2.C42 [×8], C2×C42, C2×C22⋊C4 [×15], C2×C4⋊C4 [×8], C22×D4 [×3], C23.63C23, C24.C22, C24.3C22, C232D4, C23.10D4 [×5], C23.Q8 [×2], C23.11D4 [×2], C23.81C23, C23.4Q8, C23.729C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×2], C24, C2×C4○D4, 2+ 1+4 [×5], 2- 1+4, C22.32C24, C22.34C24, C22.36C24, C22.54C24 [×2], C22.56C24 [×2], C23.729C24

Character table of C23.729C24

 class 12A2B2C2D2E2F2G2H2I2J4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O
 size 11111111888444444888888888
ρ111111111111111111111111111    trivial
ρ211111111111-1-111-1-111-1-1-1-1-1-11    linear of order 2
ρ311111111-1-1-1-1-111-1-1-111111-1-11    linear of order 2
ρ411111111-1-1-1111111-11-1-1-1-1111    linear of order 2
ρ51111111111-111-1-1-1-1-1-11-1-111-11    linear of order 2
ρ61111111111-1-1-1-1-111-1-1-111-1-111    linear of order 2
ρ711111111-1-11-1-1-1-1111-11-1-11-111    linear of order 2
ρ811111111-1-1111-1-1-1-11-1-111-11-11    linear of order 2
ρ911111111-111-1-1-1-111-111-11-11-1-1    linear of order 2
ρ1011111111-11111-1-1-1-1-11-11-11-11-1    linear of order 2
ρ11111111111-1-111-1-1-1-1111-11-1-11-1    linear of order 2
ρ12111111111-1-1-1-1-1-11111-11-111-1-1    linear of order 2
ρ1311111111-11-1-1-111-1-11-111-1-111-1    linear of order 2
ρ1411111111-11-11111111-1-1-111-1-1-1    linear of order 2
ρ15111111111-11111111-1-111-1-1-1-1-1    linear of order 2
ρ16111111111-11-1-111-1-1-1-1-1-11111-1    linear of order 2
ρ172-22-22-22-2000-2i2i-222i-2i000000000    complex lifted from C4○D4
ρ182-22-22-22-20002i-2i2-22i-2i000000000    complex lifted from C4○D4
ρ192-22-22-22-20002i-2i-22-2i2i000000000    complex lifted from C4○D4
ρ202-22-22-22-2000-2i2i2-2-2i2i000000000    complex lifted from C4○D4
ρ214-4-4-444-44000000000000000000    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
ρ2544-4-4-4-444000000000000000000    orthogonal lifted from 2+ 1+4
ρ264-4-44-444-4000000000000000000    symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of C23.729C24
On 64 points
Generators in S64
(1 45)(2 46)(3 47)(4 48)(5 63)(6 64)(7 61)(8 62)(9 23)(10 24)(11 21)(12 22)(13 20)(14 17)(15 18)(16 19)(25 30)(26 31)(27 32)(28 29)(33 38)(34 39)(35 40)(36 37)(41 49)(42 50)(43 51)(44 52)(53 60)(54 57)(55 58)(56 59)
(1 58)(2 59)(3 60)(4 57)(5 52)(6 49)(7 50)(8 51)(9 34)(10 35)(11 36)(12 33)(13 31)(14 32)(15 29)(16 30)(17 27)(18 28)(19 25)(20 26)(21 37)(22 38)(23 39)(24 40)(41 64)(42 61)(43 62)(44 63)(45 55)(46 56)(47 53)(48 54)
(1 47)(2 48)(3 45)(4 46)(5 61)(6 62)(7 63)(8 64)(9 21)(10 22)(11 23)(12 24)(13 18)(14 19)(15 20)(16 17)(25 32)(26 29)(27 30)(28 31)(33 40)(34 37)(35 38)(36 39)(41 51)(42 52)(43 49)(44 50)(53 58)(54 59)(55 60)(56 57)
(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)
(2 46)(4 48)(5 61)(6 8)(7 63)(9 36)(10 38)(11 34)(12 40)(13 26)(14 32)(15 28)(16 30)(17 27)(18 29)(19 25)(20 31)(21 39)(22 35)(23 37)(24 33)(41 43)(42 52)(44 50)(49 51)(54 57)(56 59)(62 64)
(1 52)(2 6)(3 50)(4 8)(5 58)(7 60)(9 18)(10 25)(11 20)(12 27)(13 21)(14 38)(15 23)(16 40)(17 33)(19 35)(22 32)(24 30)(26 36)(28 34)(29 39)(31 37)(41 56)(42 47)(43 54)(44 45)(46 64)(48 62)(49 59)(51 57)(53 61)(55 63)
(1 20 45 13)(2 17 46 14)(3 18 47 15)(4 19 48 16)(5 37 63 36)(6 38 64 33)(7 39 61 34)(8 40 62 35)(9 50 23 42)(10 51 24 43)(11 52 21 44)(12 49 22 41)(25 54 30 57)(26 55 31 58)(27 56 32 59)(28 53 29 60)

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

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

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

Matrix representation of C23.729C24 in GL10(𝔽5)

1000000000
0100000000
0010000000
0001000000
0000100000
0000010000
0000004000
0000000400
0000000040
0000000004
,
1000000000
0100000000
0040000000
0004000000
0000400000
0000040000
0000004000
0000000400
0000000040
0000000004
,
4000000000
0400000000
0040000000
0004000000
0000400000
0000040000
0000001000
0000000100
0000000010
0000000001
,
3000000000
0300000000
0011120000
0000400000
0001000000
0044040000
0000000130
0000004121
0000000313
0000001343
,
1000000000
0400000000
0010000000
0004000000
0000400000
0001110000
0000001000
0000000100
0000000140
0000002104
,
0100000000
1000000000
0000100000
0044430000
0010000000
0000010000
0000004000
0000000100
0000000010
0000000314
,
1000000000
0100000000
0001000000
0010000000
0044430000
0000010000
0000000100
0000004000
0000000242
0000001141

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

C23.729C24 in GAP, Magma, Sage, TeX

C_2^3._{729}C_2^4
% in TeX

G:=Group("C2^3.729C2^4");
// GroupNames label

G:=SmallGroup(128,1561);
// by ID

G=gap.SmallGroup(128,1561);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,758,723,794,185,80]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=1,d^2=c*a=a*c,g^2=a,a*b=b*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

Export

Character table of C23.729C24 in TeX

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