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G = C24.460C23  order 128 = 27

300th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.460C23, C23.714C24, C22.4872+ (1+4), C22.3722- (1+4), C23⋊Q8⋊60C2, C23.224(C2×D4), (C22×C4).410D4, C23.Q8⋊92C2, C2.61(C23⋊3D4), (C23×C4).499C22, (C22×C4).225C23, C23.7Q8⋊114C2, C22.446(C22×D4), C23.11D4⋊127C2, C23.10D4.71C2, (C22×D4).292C22, (C22×Q8).231C22, C23.78C23⋊62C2, C23.81C23⋊132C2, C2.C42.417C22, C2.66(C23.38C23), C2.57(C22.31C24), C2.57(C22.57C24), C2.49(C22.56C24), (C2×C4).431(C2×D4), (C2×C22⋊Q8)⋊47C2, (C2×C4⋊C4).523C22, (C2×C22⋊C4).333C22, (C2×C22.D4).31C2, SmallGroup(128,1546)

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

Derived series
C1 — C23 — C24.460C23
Chief series
C1 — C2 — C22 — C23 — C24 — C22×D4 — C23.10D4 — C24.460C23
Lower central
C1 — C23 — C24.460C23
Upper central
C1 — C23 — C24.460C23
Jennings
C1 — C23 — C24.460C23

Subgroups: 500 in 239 conjugacy classes, 92 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×3], C4 [×15], C22 [×3], C22 [×4], C22 [×17], C2×C4 [×4], C2×C4 [×41], D4 [×4], Q8 [×4], C23, C23 [×2], C23 [×13], C22⋊C4 [×16], C4⋊C4 [×16], C22×C4 [×5], C22×C4 [×12], C22×C4 [×2], C2×D4 [×4], C2×Q8 [×4], C24 [×2], C2.C42 [×2], C2.C42 [×8], C2×C22⋊C4 [×3], C2×C22⋊C4 [×8], C2×C4⋊C4 [×3], C2×C4⋊C4 [×8], C22⋊Q8 [×4], C22.D4 [×4], C23×C4, C22×D4, C22×Q8, C23.7Q8, C23⋊Q8, C23.10D4 [×2], C23.78C23, C23.Q8 [×2], C23.11D4, C23.11D4 [×2], C23.81C23, C23.81C23 [×2], C2×C22⋊Q8, C2×C22.D4, C24.460C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ (1+4) [×3], 2- (1+4) [×3], C23⋊3D4, C23.38C23, C22.31C24, C22.56C24 [×2], C22.57C24 [×2], C24.460C23

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

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

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

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

Matrix representation ►G ⊆ GL10(𝔽5)

4000000000
0400000000
0001000000
0010000000
0000010000
0000100000
0000001000
0000003400
0000000010
0000000024
,
1000000000
0100000000
0040000000
0004000000
0000400000
0000040000
0000001000
0000000100
0000000010
0000000001
,
1000000000
0100000000
0040000000
0004000000
0000400000
0000040000
0000004000
0000000400
0000000040
0000000004
,
4000000000
0400000000
0040000000
0004000000
0000400000
0000040000
0000001000
0000000100
0000000010
0000000001
,
4000000000
0400000000
0000100000
0000010000
0040000000
0004000000
0000004030
0000000443
0000000010
0000000001
,
0100000000
1000000000
0003000000
0030000000
0000020000
0000200000
0000002200
0000001300
0000000032
0000000012
,
1000000000
0400000000
0010000000
0004000000
0000400000
0000010000
0000001000
0000000100
0000004040
0000003404

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

Character table of C24.460C23

 class 12A2B2C2D2E2F2G2H2I2J4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O
 size 11111111448444488888888888
ρ111111111111111111111111111    trivial
ρ211111111-1-11-1-111-1-11-1-111-11-11    linear of order 2
ρ31111111111-11111-11-1-1-11-1-111-1    linear of order 2
ρ411111111-1-1-1-1-1111-1-1111-111-1-1    linear of order 2
ρ511111111-1-1-1-1-111-111-11-1-11-111    linear of order 2
ρ61111111111-111111-111-1-1-1-1-1-11    linear of order 2
ρ711111111-1-11-1-11111-11-1-11-1-11-1    linear of order 2
ρ8111111111111111-1-1-1-11-111-1-1-1    linear of order 2
ρ911111111-1-1-111-1-1-1-1-11111-1-111    linear of order 2
ρ101111111111-1-1-1-1-111-1-1-1111-1-11    linear of order 2
ρ1111111111-1-1111-1-11-11-1-11-11-11-1    linear of order 2
ρ1211111111111-1-1-1-1-111111-1-1-1-1-1    linear of order 2
ρ1311111111111-1-1-1-11-1-1-11-1-1-1111    linear of order 2
ρ1411111111-1-1111-1-1-11-11-1-1-111-11    linear of order 2
ρ151111111111-1-1-1-1-1-1-111-1-11111-1    linear of order 2
ρ1611111111-1-1-111-1-1111-11-11-11-1-1    linear of order 2
ρ172-22-22-22-22-202-22-200000000000    orthogonal lifted from D4
ρ182-22-22-22-22-20-22-2200000000000    orthogonal lifted from D4
ρ192-22-22-22-2-2202-2-2200000000000    orthogonal lifted from D4
ρ202-22-22-22-2-220-222-200000000000    orthogonal lifted from D4
ρ2144-4-44-4-44000000000000000000    orthogonal lifted from 2+ (1+4)
ρ224-44-4-44-44000000000000000000    orthogonal lifted from 2+ (1+4)
ρ234-4-4444-4-4000000000000000000    orthogonal lifted from 2+ (1+4)
ρ244-4-44-4-444000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ2544-4-4-444-4000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ264444-4-4-4-4000000000000000000    symplectic lifted from 2- (1+4), Schur index 2

In GAP, Magma, Sage, TeX 

C_2^4._{460}C_2^3
 
% in TeX
 
G:=Group("C2^4.460C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1546);
 
// by ID
 
G=gap.SmallGroup(128,1546);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,794,185]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=g^2=1,e^2=f^2=b,g*a*g=a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f^-1=a*b*c,b*c=c*b,b*d=d*b,f*e*f^-1=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,g*e*g=c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,g*f*g=d*f=f*d,d*g=g*d>;
 
// generators/relations
 

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