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G = C23.724C24order 128 = 27

441st central stem extension by C23 of C24

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

Aliases: C23.724C24, C24.106C23, C22.4972+ 1+4, C22.3802- 1+4, C23⋊Q863C2, C232D4.36C2, C23.104(C4○D4), (C23×C4).180C22, (C22×C4).235C23, C23.8Q8142C2, C23.23D4108C2, C23.11D4130C2, C23.10D4111C2, (C22×D4).299C22, (C22×Q8).237C22, C2.17(C24⋊C22), C23.83C23133C2, C2.47(C22.54C24), C2.113(C22.32C24), C2.C42.427C22, C2.53(C22.56C24), C2.122(C22.33C24), (C2×C4⋊C4).533C22, C22.572(C2×C4○D4), (C2×C22⋊C4).342C22, SmallGroup(128,1556)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.724C24
C1C2C22C23C24C22×D4C232D4 — C23.724C24
C1C23 — C23.724C24
C1C23 — C23.724C24
C1C23 — C23.724C24

Generators and relations for C23.724C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=g2=1, d2=c, ab=ba, ac=ca, ede=ad=da, ae=ea, gfg=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=abe >

Subgroups: 564 in 236 conjugacy classes, 84 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×12], C22 [×3], C22 [×4], C22 [×24], C2×C4 [×40], D4 [×12], Q8 [×4], C23, C23 [×2], C23 [×20], C22⋊C4 [×15], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×10], C22×C4 [×3], C2×D4 [×11], C2×Q8 [×3], C24, C24 [×2], C2.C42 [×2], C2.C42 [×10], C2×C22⋊C4 [×2], C2×C22⋊C4 [×12], C2×C4⋊C4 [×4], C23×C4, C22×D4, C22×D4 [×2], C22×Q8, C23.8Q8, C23.23D4 [×2], C232D4, C23⋊Q8, C23⋊Q8 [×2], C23.10D4 [×4], C23.11D4 [×2], C23.83C23 [×2], C23.724C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×2], C24, C2×C4○D4, 2+ 1+4 [×5], 2- 1+4, C22.32C24 [×2], C22.33C24, C22.54C24, C24⋊C22, C22.56C24 [×2], C23.724C24

Character table of C23.724C24

 class 12A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L4M4N
 size 11111111448844448888888888
ρ111111111111111111111111111    trivial
ρ211111111-1-1-11-1-111-111-1-111-1-11    linear of order 2
ρ31111111111-1-1-1-1-1-111-1-11111-1-1    linear of order 2
ρ411111111-1-11-111-1-1-11-11-111-11-1    linear of order 2
ρ511111111111-1-1-1-1-1-1-11-1-1-11111    linear of order 2
ρ611111111-1-1-1-111-1-11-1111-11-1-11    linear of order 2
ρ71111111111-111111-1-1-11-1-111-1-1    linear of order 2
ρ811111111-1-111-1-1111-1-1-11-11-11-1    linear of order 2
ρ911111111-1-11-1-1-111-11-111-1-11-11    linear of order 2
ρ101111111111-1-1111111-1-1-1-1-1-111    linear of order 2
ρ1111111111-1-1-1111-1-1-111-11-1-111-1    linear of order 2
ρ12111111111111-1-1-1-11111-1-1-1-1-1-1    linear of order 2
ρ1311111111-1-11111-1-11-1-1-1-11-11-11    linear of order 2
ρ141111111111-11-1-1-1-1-1-1-1111-1-111    linear of order 2
ρ1511111111-1-1-1-1-1-1111-111-11-111-1    linear of order 2
ρ1611111111111-11111-1-11-111-1-1-1-1    linear of order 2
ρ172-22-22-22-22-2002i-2i2i-2i0000000000    complex lifted from C4○D4
ρ182-22-22-22-22-200-2i2i-2i2i0000000000    complex lifted from C4○D4
ρ192-22-22-22-2-2200-2i2i2i-2i0000000000    complex lifted from C4○D4
ρ202-22-22-22-2-22002i-2i-2i2i0000000000    complex lifted from C4○D4
ρ2144-4-44-4-44000000000000000000    orthogonal lifted from 2+ 1+4
ρ224-4-44-4-444000000000000000000    orthogonal lifted from 2+ 1+4
ρ234-4-4444-4-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ2444-4-4-444-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ254444-4-4-4-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ264-44-4-44-44000000000000000000    symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of C23.724C24
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)
(2 10)(4 12)(5 64)(6 35)(7 62)(8 33)(13 55)(14 28)(15 53)(16 26)(17 19)(18 48)(20 46)(22 50)(24 52)(25 43)(27 41)(29 31)(30 60)(32 58)(34 38)(36 40)(37 63)(39 61)(42 56)(44 54)(45 47)(57 59)
(1 45)(2 60)(3 47)(4 58)(5 54)(6 41)(7 56)(8 43)(9 17)(10 32)(11 19)(12 30)(13 39)(14 34)(15 37)(16 36)(18 24)(20 22)(21 29)(23 31)(25 35)(26 38)(27 33)(28 40)(42 64)(44 62)(46 52)(48 50)(49 57)(51 59)(53 61)(55 63)
(1 41)(2 42)(3 43)(4 44)(5 20)(6 17)(7 18)(8 19)(9 13)(10 14)(11 15)(12 16)(21 25)(22 26)(23 27)(24 28)(29 61)(30 62)(31 63)(32 64)(33 59)(34 60)(35 57)(36 58)(37 47)(38 48)(39 45)(40 46)(49 53)(50 54)(51 55)(52 56)

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

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

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

Matrix representation of C23.724C24 in GL10(𝔽5)

1000000000
0100000000
0040000000
0004000000
0000400000
0000040000
0000004000
0000000400
0000000040
0000000004
,
1000000000
0100000000
0040000000
0004000000
0000400000
0000040000
0000001000
0000000100
0000000010
0000000001
,
4000000000
0400000000
0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
2000000000
0200000000
0000100000
0011430000
0010000000
0000040000
0000000010
0000001144
0000001000
0000000004
,
1000000000
3400000000
0010000000
0001000000
0000400000
0011040000
0000001000
0000000400
0000000040
0000000321
,
2200000000
1300000000
0010000000
0004000000
0000400000
0004110000
0000001000
0000000400
0000000010
0000002034
,
4000000000
0400000000
0001000000
0010000000
0011430000
0000010000
0000000100
0000001000
0000001144
0000000001

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

C23.724C24 in GAP, Magma, Sage, TeX

C_2^3._{724}C_2^4
% in TeX

G:=Group("C2^3.724C2^4");
// GroupNames label

G:=SmallGroup(128,1556);
// by ID

G=gap.SmallGroup(128,1556);
# by ID

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

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

Export

Character table of C23.724C24 in TeX

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