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G = C23.736C24  order 128 = 27

453rd central stem extension by C23 of C24

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

Aliases: C24.115C23, C23.736C24, C22.3902- (1+4), C22.5092+ (1+4), (C22×C4).247C23, (C2×C42).742C22, C23.Q8.45C2, C23.4Q8.33C2, C23.11D4.64C2, C24.C22.83C2, C23.81C23⋊141C2, C23.83C23⋊140C2, C23.65C23⋊164C2, C23.63C23⋊200C2, C2.C42.439C22, C2.60(C22.56C24), C2.70(C22.57C24), C2.66(C22.35C24), C2.70(C22.34C24), C2.127(C22.33C24), (C2×C4).258(C4○D4), (C2×C4⋊C4).545C22, C22.584(C2×C4○D4), (C2×C22⋊C4).353C22, SmallGroup(128,1568)

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

Derived series
C1 — C23 — C23.736C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C23.4Q8 — C23.736C24
Lower central
C1 — C23 — C23.736C24
Upper central
C1 — C23 — C23.736C24
Jennings
C1 — C23 — C23.736C24

Subgroups: 356 in 179 conjugacy classes, 84 normal (82 characteristic)
C1, C2 [×7], C2, C4 [×15], C22 [×7], C22 [×7], C2×C4 [×2], C2×C4 [×41], C23, C23 [×7], C42, C22⋊C4 [×8], C4⋊C4 [×15], C22×C4 [×14], C24, C2.C42 [×14], C2×C42, C2×C22⋊C4 [×7], C2×C4⋊C4 [×13], C23.63C23, C24.C22, C23.65C23, C23.Q8 [×2], C23.11D4 [×3], C23.81C23 [×4], C23.4Q8, C23.83C23 [×2], C23.736C24

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, C22.34C24, C22.35C24, C22.56C24 [×2], C22.57C24 [×2], C23.736C24

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

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

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

Matrix representation ►G ⊆ GL10(𝔽5)

1000000000
0100000000
0040000000
0004000000
0000400000
0000040000
0000004000
0000000400
0000000040
0000000004
,
1000000000
0100000000
0010000000
0001000000
0000100000
0000010000
0000004000
0000000400
0000000040
0000000004
,
4000000000
0400000000
0040000000
0004000000
0000400000
0000040000
0000001000
0000000100
0000000010
0000000001
,
3000000000
0300000000
0020300000
0000110000
0000300000
0004200000
0000000010
0000000001
0000001000
0000000100
,
1000000000
0400000000
0010000000
0044000000
0020400000
0000010000
0000001000
0000000100
0000000040
0000000004
,
0100000000
1000000000
0043000000
0001000000
0003010000
0002100000
0000003000
0000000200
0000000020
0000000003
,
1000000000
0100000000
0043000000
0011000000
0003010000
0022400000
0000000100
0000004000
0000000001
0000000040

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

Character table of C23.736C24

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q
 size 11111111844444488888888888
ρ111111111111111111111111111    trivial
ρ211111111-1-1-1-1-11111-11-111-1-11-1    linear of order 2
ρ311111111-1-1-111-1-1-1-111-1-11111-1    linear of order 2
ρ411111111111-1-1-1-1-1-1-111-11-1-111    linear of order 2
ρ511111111111-1-1-1-11-11-1-11-11-11-1    linear of order 2
ρ611111111-1-1-111-1-11-1-1-111-1-1111    linear of order 2
ρ711111111-1-1-1-1-111-111-11-1-11-111    linear of order 2
ρ8111111111111111-11-1-1-1-1-1-111-1    linear of order 2
ρ911111111-1111111-1-1-1-11111-1-1-1    linear of order 2
ρ10111111111-1-1-1-111-1-11-1-111-11-11    linear of order 2
ρ11111111111-1-111-1-111-1-1-1-111-1-11    linear of order 2
ρ1211111111-111-1-1-1-1111-11-11-11-1-1    linear of order 2
ρ1311111111-111-1-1-1-1-11-11-11-111-11    linear of order 2
ρ14111111111-1-111-1-1-111111-1-1-1-1-1    linear of order 2
ρ15111111111-1-1-1-1111-1-111-1-111-1-1    linear of order 2
ρ1611111111-11111111-111-1-1-1-1-1-11    linear of order 2
ρ172-22-22-22-202i2i2-22i2i00000000000    complex lifted from C4○D4
ρ182-22-22-22-202i2i2-22i2i00000000000    complex lifted from C4○D4
ρ192-22-22-22-202i2i-222i2i00000000000    complex lifted from C4○D4
ρ202-22-22-22-202i2i-222i2i00000000000    complex lifted from C4○D4
ρ214-4-4-444-44000000000000000000    orthogonal lifted from 2+ (1+4)
ρ224-444-4-4-44000000000000000000    orthogonal lifted from 2+ (1+4)
ρ23444-4-44-4-4000000000000000000    orthogonal lifted from 2+ (1+4)
ρ2444-444-4-4-4000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ254-4-44-444-4000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ2644-4-4-4-444000000000000000000    symplectic lifted from 2- (1+4), Schur index 2

In GAP, Magma, Sage, TeX 

C_2^3._{736}C_2^4
 
% in TeX
 
G:=Group("C2^3.736C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1568);
 
// by ID
 
G=gap.SmallGroup(128,1568);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,723,184,794,185,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=1,d^2=c,f^2=b,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^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=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
 

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