Copied to
clipboard

G = C23.715C24order 128 = 27

432nd central stem extension by C23 of C24

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

Aliases: C24.100C23, C23.715C24, C22.4882+ 1+4, (C22×C4)⋊41D4, C232D447C2, C23.225(C2×D4), C23.34D461C2, C2.62(C233D4), (C23×C4).500C22, (C22×C4).226C23, C22.447(C22×D4), C23.11D4128C2, (C22×D4).293C22, C2.68(C22.29C24), C2.44(C22.54C24), C2.C42.418C22, (C2×C4⋊D4)⋊40C2, (C2×C4).432(C2×D4), (C2×C4⋊C4).524C22, (C2×C22⋊C4).334C22, SmallGroup(128,1547)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.715C24
C1C2C22C23C22×C4C22×D4C232D4 — C23.715C24
C1C23 — C23.715C24
C1C23 — C23.715C24
C1C23 — C23.715C24

Generators and relations for C23.715C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=1, d2=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 >

Subgroups: 820 in 322 conjugacy classes, 92 normal (8 characteristic)
C1, C2, C2 [×6], C2 [×6], C4 [×12], C22, C22 [×6], C22 [×38], C2×C4 [×4], C2×C4 [×32], D4 [×40], C23, C23 [×2], C23 [×34], C22⋊C4 [×18], C4⋊C4 [×4], C22×C4 [×14], C22×C4 [×2], C2×D4 [×36], C24, C24 [×4], C2.C42 [×8], C2×C22⋊C4 [×14], C2×C4⋊C4 [×2], C4⋊D4 [×8], C23×C4, C22×D4 [×10], C23.34D4, C232D4 [×8], C23.11D4 [×4], C2×C4⋊D4 [×2], C23.715C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ 1+4 [×6], C233D4, C22.29C24 [×2], C22.54C24 [×4], C23.715C24

Character table of C23.715C24

 class 12A2B2C2D2E2F2G2H2I2J2K2L2M4A4B4C4D4E4F4G4H4I4J4K4L
 size 11111111448888444488888888
ρ111111111111111111111111111    trivial
ρ211111111-1-1-11111-11-11-1-1-11-11-1    linear of order 2
ρ3111111111111-1-11111-11-1-1-1-11-1    linear of order 2
ρ411111111-1-1-11-1-11-11-1-1-111-1111    linear of order 2
ρ51111111111-1-1-1-111111-1111-1-1-1    linear of order 2
ρ611111111-1-11-1-1-11-11-111-1-111-11    linear of order 2
ρ71111111111-1-1111111-1-1-1-1-11-11    linear of order 2
ρ811111111-1-11-1111-11-1-1111-1-1-1-1    linear of order 2
ρ91111111111-11-11-1-1-1-1-111-11-1-11    linear of order 2
ρ1011111111-1-111-11-11-11-1-1-1111-1-1    linear of order 2
ρ111111111111-111-1-1-1-1-111-11-11-1-1    linear of order 2
ρ1211111111-1-1111-1-11-111-11-1-1-1-11    linear of order 2
ρ1311111111111-11-1-1-1-1-1-1-11-1111-1    linear of order 2
ρ1411111111-1-1-1-11-1-11-11-11-111-111    linear of order 2
ρ1511111111111-1-11-1-1-1-11-1-11-1-111    linear of order 2
ρ1611111111-1-1-1-1-11-11-11111-1-111-1    linear of order 2
ρ172-22-22-22-22-200002-2-2200000000    orthogonal lifted from D4
ρ182-22-22-22-2-22000022-2-200000000    orthogonal lifted from D4
ρ192-22-22-22-2-220000-2-22200000000    orthogonal lifted from D4
ρ202-22-22-22-22-20000-222-200000000    orthogonal lifted from 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

Smallest permutation representation of C23.715C24
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 41)(2 42)(3 43)(4 44)(5 20)(6 17)(7 18)(8 19)(9 57)(10 58)(11 59)(12 60)(13 25)(14 26)(15 27)(16 28)(21 34)(22 35)(23 36)(24 33)(29 45)(30 46)(31 47)(32 48)(37 56)(38 53)(39 54)(40 55)(49 61)(50 62)(51 63)(52 64)
(1 11)(2 12)(3 9)(4 10)(5 50)(6 51)(7 52)(8 49)(13 31)(14 32)(15 29)(16 30)(17 63)(18 64)(19 61)(20 62)(21 40)(22 37)(23 38)(24 39)(25 47)(26 48)(27 45)(28 46)(33 54)(34 55)(35 56)(36 53)(41 59)(42 60)(43 57)(44 58)
(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 61)(2 64)(3 63)(4 62)(5 58)(6 57)(7 60)(8 59)(9 17)(10 20)(11 19)(12 18)(13 21)(14 24)(15 23)(16 22)(25 34)(26 33)(27 36)(28 35)(29 38)(30 37)(31 40)(32 39)(41 49)(42 52)(43 51)(44 50)(45 53)(46 56)(47 55)(48 54)
(1 25)(2 14)(3 27)(4 16)(5 35)(6 23)(7 33)(8 21)(9 45)(10 30)(11 47)(12 32)(13 41)(15 43)(17 36)(18 24)(19 34)(20 22)(26 42)(28 44)(29 57)(31 59)(37 62)(38 51)(39 64)(40 49)(46 58)(48 60)(50 56)(52 54)(53 63)(55 61)
(1 27 3 25)(2 28 4 26)(5 56 7 54)(6 53 8 55)(9 47 11 45)(10 48 12 46)(13 41 15 43)(14 42 16 44)(17 38 19 40)(18 39 20 37)(21 63 23 61)(22 64 24 62)(29 57 31 59)(30 58 32 60)(33 50 35 52)(34 51 36 49)

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

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

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

Matrix representation of C23.715C24 in GL10(𝔽5)

1000000000
0100000000
0010000000
0001000000
0000100000
0000010000
0000004000
0000000400
0000000040
0000000004
,
1000000000
0100000000
0040000000
0004000000
0000400000
0000040000
0000001000
0000000100
0000000010
0000000001
,
4000000000
0400000000
0040000000
0004000000
0000400000
0000040000
0000001000
0000000100
0000000010
0000000001
,
4000000000
0400000000
0000100000
0000010000
0010000000
0001000000
0000003000
0000000300
0000000020
0000000002
,
1300000000
0400000000
0001000000
0010000000
0000010000
0000100000
0000000010
0000000001
0000001000
0000000100
,
4000000000
4100000000
0010000000
0004000000
0000400000
0000010000
0000001200
0000000400
0000000012
0000000004
,
4000000000
0400000000
0010000000
0004000000
0000100000
0000040000
0000001200
0000004400
0000000043
0000000011

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],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,2],[1,0,0,0,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[4,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,4],[4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,0,2,4,0,0,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,0,0,3,1] >;

C23.715C24 in GAP, Magma, Sage, TeX

C_2^3._{715}C_2^4
% in TeX

G:=Group("C2^3.715C2^4");
// GroupNames label

G:=SmallGroup(128,1547);
// by ID

G=gap.SmallGroup(128,1547);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,758,723,794,185,80]);
// Polycyclic

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

Export

Character table of C23.715C24 in TeX

׿
×
𝔽