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G = C23.730C24order 128 = 27

447th central stem extension by C23 of C24

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

Aliases: C24.112C23, C23.730C24, C22.5032+ 1+4, C22.3842- 1+4, C23⋊Q864C2, C23.Q899C2, (C2×C42).738C22, (C22×C4).241C23, C23.11D4136C2, C23.10D4.75C2, (C22×D4).305C22, (C22×Q8).238C22, C23.78C2366C2, C24.3C22.80C2, C23.63C23198C2, C23.81C23138C2, C2.119(C22.32C24), C2.53(C22.54C24), C2.C42.433C22, C2.64(C22.57C24), C2.57(C22.56C24), C2.125(C22.36C24), (C2×C4).254(C4○D4), (C2×C4⋊C4).539C22, C22.578(C2×C4○D4), (C2×C22⋊C4).348C22, SmallGroup(128,1562)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.730C24
C1C2C22C23C22×C4C22×D4C23.10D4 — C23.730C24
C1C23 — C23.730C24
C1C23 — C23.730C24
C1C23 — C23.730C24

Generators and relations for C23.730C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=c, e2=g2=a, ab=ba, ac=ca, ede-1=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: 452 in 204 conjugacy classes, 84 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×14], C22 [×3], C22 [×4], C22 [×14], C2×C4 [×2], C2×C4 [×38], D4 [×4], Q8 [×4], C23, C23 [×14], C42, C22⋊C4 [×12], C4⋊C4 [×10], C22×C4 [×3], C22×C4 [×10], C2×D4 [×4], C2×Q8 [×3], C24 [×2], C2.C42 [×2], C2.C42 [×10], C2×C42, C2×C22⋊C4 [×12], C2×C4⋊C4 [×2], C2×C4⋊C4 [×6], C22×D4, C22×Q8, C23.63C23 [×2], C24.3C22, C23⋊Q8 [×2], C23.10D4 [×2], C23.78C23, C23.Q8 [×2], C23.11D4 [×4], C23.81C23, C23.730C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×2], C24, C2×C4○D4, 2+ 1+4 [×4], 2- 1+4 [×2], C22.32C24, C22.36C24 [×2], C22.54C24, C22.56C24 [×2], C22.57C24, C23.730C24

Character table of C23.730C24

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P
 size 11111111884444448888888888
ρ111111111111111111111111111    trivial
ρ211111111-1-1-1-111-1-1-1-11111-1-111    linear of order 2
ρ311111111-1-1111111-1-111-1-111-1-1    linear of order 2
ρ41111111111-1-111-1-11111-1-1-1-1-1-1    linear of order 2
ρ511111111-1-1-1-111-1-111-1-11111-1-1    linear of order 2
ρ61111111111111111-1-1-1-111-1-1-1-1    linear of order 2
ρ71111111111-1-111-1-1-1-1-1-1-1-11111    linear of order 2
ρ811111111-1-111111111-1-1-1-1-1-111    linear of order 2
ρ911111111-1111-1-1-1-1-11-11-111-11-1    linear of order 2
ρ10111111111-1-1-1-1-1111-1-11-11-111-1    linear of order 2
ρ11111111111-111-1-1-1-11-1-111-11-1-11    linear of order 2
ρ1211111111-11-1-1-1-111-11-111-1-11-11    linear of order 2
ρ13111111111-1-1-1-1-111-111-1-111-1-11    linear of order 2
ρ1411111111-1111-1-1-1-11-11-1-11-11-11    linear of order 2
ρ1511111111-11-1-1-1-1111-11-11-11-11-1    linear of order 2
ρ16111111111-111-1-1-1-1-111-11-1-111-1    linear of order 2
ρ172-22-22-22-200-2i2i2-2-2i2i0000000000    complex lifted from C4○D4
ρ182-22-22-22-200-2i2i-222i-2i0000000000    complex lifted from C4○D4
ρ192-22-22-22-2002i-2i-22-2i2i0000000000    complex lifted from C4○D4
ρ202-22-22-22-2002i-2i2-22i-2i0000000000    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
ρ244-4-4-444-44000000000000000000    orthogonal lifted from 2+ 1+4
ρ25444-4-44-4-4000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ2644-4-4-4-444000000000000000000    symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of C23.730C24
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 17 9 45)(2 46 10 18)(3 19 11 47)(4 48 12 20)(5 16 38 44)(6 41 39 13)(7 14 40 42)(8 43 37 15)(21 57 49 29)(22 30 50 58)(23 59 51 31)(24 32 52 60)(25 61 53 35)(26 36 54 62)(27 63 55 33)(28 34 56 64)
(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), (1,17,9,45)(2,46,10,18)(3,19,11,47)(4,48,12,20)(5,16,38,44)(6,41,39,13)(7,14,40,42)(8,43,37,15)(21,57,49,29)(22,30,50,58)(23,59,51,31)(24,32,52,60)(25,61,53,35)(26,36,54,62)(27,63,55,33)(28,34,56,64), (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), (1,17,9,45)(2,46,10,18)(3,19,11,47)(4,48,12,20)(5,16,38,44)(6,41,39,13)(7,14,40,42)(8,43,37,15)(21,57,49,29)(22,30,50,58)(23,59,51,31)(24,32,52,60)(25,61,53,35)(26,36,54,62)(27,63,55,33)(28,34,56,64), (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)], [(1,17,9,45),(2,46,10,18),(3,19,11,47),(4,48,12,20),(5,16,38,44),(6,41,39,13),(7,14,40,42),(8,43,37,15),(21,57,49,29),(22,30,50,58),(23,59,51,31),(24,32,52,60),(25,61,53,35),(26,36,54,62),(27,63,55,33),(28,34,56,64)], [(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 of C23.730C24 in GL10(𝔽5)

1000000000
0100000000
0040000000
0004000000
0000400000
0000040000
0000001000
0000000100
0000000010
0000000001
,
1000000000
0100000000
0010000000
0001000000
0000100000
0000010000
0000004000
0000000400
0000000040
0000000004
,
4000000000
0400000000
0040000000
0004000000
0000400000
0000040000
0000004000
0000000400
0000000040
0000000004
,
3000000000
0300000000
0030000000
0003000000
0000200000
0000020000
0000002030
0000001303
0000000030
0000000042
,
0100000000
1000000000
0000100000
0000010000
0040000000
0004000000
0000001000
0000000400
0000000010
0000001004
,
1000000000
0400000000
0001000000
0010000000
0000040000
0000400000
0000000100
0000001000
0000003004
0000000340
,
1000000000
0100000000
0001000000
0040000000
0000040000
0000100000
0000000100
0000001000
0000002201
0000003310

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

C23.730C24 in GAP, Magma, Sage, TeX

C_2^3._{730}C_2^4
% in TeX

G:=Group("C2^3.730C2^4");
// GroupNames label

G:=SmallGroup(128,1562);
// by ID

G=gap.SmallGroup(128,1562);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,723,794,185,80]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=c,e^2=g^2=a,a*b=b*a,a*c=c*a,e*d*e^-1=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 C23.730C24 in TeX

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