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G = C24.464C23  order 128 = 27

304th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.464C23, C23.735C24, C22.5082+ (1+4), C22.3892- (1+4), C23⋊Q8.33C2, C23.109(C4○D4), (C23×C4).185C22, (C22×C4).246C23, C23.8Q8.69C2, C23.11D4.63C2, (C22×Q8).242C22, C23.78C23⋊69C2, C2.20(C24⋊C22), C23.83C23⋊139C2, C2.C42.438C22, C2.69(C22.57C24), C2.126(C22.33C24), (C2×C4⋊C4).544C22, C22.583(C2×C4○D4), (C2×C22⋊C4).352C22, SmallGroup(128,1567)

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

Derived series
C1 — C23 — C24.464C23
Chief series
C1 — C2 — C22 — C23 — C24 — C2×C22⋊C4 — C23.11D4 — C24.464C23
Lower central
C1 — C23 — C24.464C23
Upper central
C1 — C23 — C24.464C23
Jennings
C1 — C23 — C24.464C23

Subgroups: 388 in 190 conjugacy classes, 84 normal (9 characteristic)
C1, C2, C2 [×6], C2 [×2], C4 [×14], C22, C22 [×6], C22 [×10], C2×C4 [×46], Q8 [×4], C23, C23 [×2], C23 [×6], C22⋊C4 [×9], C4⋊C4 [×9], C22×C4 [×14], C22×C4 [×3], C2×Q8 [×3], C24, C2.C42 [×18], C2×C22⋊C4 [×6], C2×C4⋊C4 [×9], C23×C4, C22×Q8, C23.8Q8 [×3], C23⋊Q8, C23.78C23 [×2], C23.11D4 [×3], C23.83C23 [×6], C24.464C23

Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×2], C24, C2×C4○D4, 2+ (1+4) [×3], 2- (1+4) [×3], C22.33C24 [×3], C24⋊C22, C22.57C24 [×3], C24.464C23

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

Smallest permutation representation
►On 64 points
Generators in S64
(5 62)(6 63)(7 64)(8 61)(13 41)(14 42)(15 43)(16 44)(17 59)(18 60)(19 57)(20 58)(25 53)(26 54)(27 55)(28 56)(29 47)(30 48)(31 45)(32 46)(33 39)(34 40)(35 37)(36 38)

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

Matrix representation ►G ⊆ GL10(𝔽5)

4000000000
0400000000
0010000000
0004000000
0004100000
0000040000
0000001000
0000000400
0000000010
0000000004
,
1000000000
0100000000
0040000000
0004000000
0000400000
0000040000
0000004000
0000000400
0000000040
0000000004
,
1000000000
0100000000
0010000000
0001000000
0000100000
0000010000
0000004000
0000000400
0000000040
0000000004
,
4000000000
0400000000
0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
3000000000
0300000000
0041300000
0000030000
0002140000
0002000000
0000004000
0000000100
0000000010
0000000004
,
0100000000
1000000000
0004000000
0010000000
0033010000
0023400000
0000000010
0000000001
0000004000
0000000400
,
4000000000
0100000000
0001000000
0040000000
0022010000
0023400000
0000000100
0000001000
0000000001
0000000010

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

Character table of C24.464C23

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P
 size 11111111444444888888888888
ρ111111111111111111111111111    trivial
ρ211111111-1-1-1-111-111-1-11-11-111-1    linear of order 2
ρ311111111111111-1-111-1-1-1-1-1-111    linear of order 2
ρ411111111-1-1-1-1111-11-11-11-11-11-1    linear of order 2
ρ511111111-1-111-1-11-1-11-111-1-111-1    linear of order 2
ρ61111111111-1-1-1-1-1-1-1-111-1-11111    linear of order 2
ρ711111111-1-111-1-1-11-111-1-111-11-1    linear of order 2
ρ81111111111-1-1-1-111-1-1-1-111-1-111    linear of order 2
ρ91111111111111111-1-111-1-1-1-1-1-1    linear of order 2
ρ1011111111-1-1-1-111-11-11-111-11-1-11    linear of order 2
ρ1111111111111111-1-1-1-1-1-11111-1-1    linear of order 2
ρ1211111111-1-1-1-1111-1-111-1-11-11-11    linear of order 2
ρ1311111111-1-111-1-11-11-1-11-111-1-11    linear of order 2
ρ141111111111-1-1-1-1-1-1111111-1-1-1-1    linear of order 2
ρ1511111111-1-111-1-1-111-11-11-1-11-11    linear of order 2
ρ161111111111-1-1-1-11111-1-1-1-111-1-1    linear of order 2
ρ17222-2-2-22-2-222i2i2i2i000000000000    complex lifted from C4○D4
ρ18222-2-2-22-22-22i2i2i2i000000000000    complex lifted from C4○D4
ρ19222-2-2-22-2-222i2i2i2i000000000000    complex lifted from C4○D4
ρ20222-2-2-22-22-22i2i2i2i000000000000    complex lifted from C4○D4
ρ214-4444-4-4-4000000000000000000    orthogonal lifted from 2+ (1+4)
ρ224-4-4-44-444000000000000000000    orthogonal lifted from 2+ (1+4)
ρ2344-4-444-4-4000000000000000000    orthogonal lifted from 2+ (1+4)
ρ2444-44-4-4-44000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ254-4-44-444-4000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ264-44-4-44-44000000000000000000    symplectic lifted from 2- (1+4), Schur index 2

In GAP, Magma, Sage, TeX 

C_2^4._{464}C_2^3
 
% in TeX
 
G:=Group("C2^4.464C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1567);
 
// by ID
 
G=gap.SmallGroup(128,1567);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,723,268,794,185]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d,f^2=b,g^2=c*b=b*c,g*a*g^-1=a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f^-1=a*b*c,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^-1=c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,g*f*g^-1=d*f=f*d,d*g=g*d>;
 
// generators/relations
 

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