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G = C23.713C24  order 128 = 27

430th central stem extension by C23 of C24

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

Aliases: C23.713C24, C24.459C23, C22.3712- (1+4), C22.4862+ (1+4), (C22×C4)⋊40D4, C23⋊2D4⋊46C2, C23.223(C2×D4), C23.Q8⋊91C2, C23.4Q8⋊64C2, C2.60(C23⋊3D4), (C22×C4).224C23, (C23×C4).498C22, C23.7Q8⋊113C2, C22.445(C22×D4), C23.10D4⋊107C2, (C22×D4).291C22, C2.67(C22.29C24), C23.83C23⋊130C2, C2.43(C22.54C24), C2.C42.416C22, C2.48(C22.56C24), C2.56(C22.31C24), (C2×C4⋊D4)⋊39C2, (C2×C4).430(C2×D4), (C2×C4⋊C4).522C22, (C2×C22.D4)⋊44C2, (C2×C22⋊C4).332C22, SmallGroup(128,1545)

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

Derived series
C1 — C23 — C23.713C24
Chief series
C1 — C2 — C22 — C23 — C24 — C22×D4 — C23⋊2D4 — C23.713C24
Lower central
C1 — C23 — C23.713C24
Upper central
C1 — C23 — C23.713C24
Jennings
C1 — C23 — C23.713C24

Subgroups: 692 in 289 conjugacy classes, 92 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×5], C4 [×13], C22 [×3], C22 [×4], C22 [×31], C2×C4 [×4], C2×C4 [×35], D4 [×24], C23, C23 [×2], C23 [×27], C22⋊C4 [×20], C4⋊C4 [×10], C22×C4 [×5], C22×C4 [×10], C22×C4 [×2], C2×D4 [×22], C24 [×2], C24 [×2], C2.C42 [×2], C2.C42 [×4], C2×C22⋊C4 [×3], C2×C22⋊C4 [×12], C2×C4⋊C4 [×3], C2×C4⋊C4 [×4], C4⋊D4 [×4], C22.D4 [×4], C23×C4, C22×D4 [×2], C22×D4 [×4], C23.7Q8, C23⋊2D4, C23⋊2D4 [×2], C23.10D4, C23.10D4 [×4], C23.Q8 [×2], C23.4Q8, C23.83C23, C2×C4⋊D4, C2×C22.D4, C23.713C24

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ (1+4) [×5], 2- (1+4), C23⋊3D4, C22.29C24, C22.31C24, C22.54C24 [×2], C22.56C24 [×2], C23.713C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=g2=1, d2=a, 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 >

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

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

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

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

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

Matrix representation ►G ⊆ GL10(ℤ)

1000000000
0100000000
00-10000000
000-1000000
0000-100000
00000-10000
000000-1000
0000000-100
00000000-10
000000000-1
,
1000000000
0100000000
0010000000
0001000000
0000100000
0000010000
000000-1000
0000000-100
00000000-10
000000000-1
,
-1000000000
0-100000000
0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
-1000000000
0-100000000
00000-10000
0000100000
000-1000000
0010000000
0000000-120
000000100-2
000000000-1
0000000010
,
-1000000000
0100000000
0000010000
0000100000
0001000000
0010000000
00000001-20
000000100-2
000000000-1
00000000-10
,
0-100000000
-1000000000
0000100000
0000010000
0010000000
0001000000
0000000100
0000001000
0000000001
0000000010
,
-1000000000
0-100000000
0001000000
0010000000
00000-10000
0000-100000
0000001000
0000000-100
0000000-110
000000100-1

G:=sub<GL(10,Integers())| [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,-1,0,0,0,0,0,0,0,0,0,0,-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,-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,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,-2,-1,0],[-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-2,0,0,-1,0,0,0,0,0,0,0,-2,-1,0],[0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1] >;
 

Character table of C23.713C24

 class 12A2B2C2D2E2F2G2H2I2J2K2L4A4B4C4D4E4F4G4H4I4J4K4L4M
 size 11111111448884444888888888
ρ111111111111111111111111111    trivial
ρ211111111-1-1-11-11-1-111-11-1-111-11    linear of order 2
ρ311111111-1-1-111-111-11-1-11-1-1-111    linear of order 2
ρ4111111111111-1-1-1-1-111-1-11-1-1-11    linear of order 2
ρ51111111111-11-1-1-1-1-1-1111-1-111-1    linear of order 2
ρ611111111-1-1111-111-1-1-11-11-11-1-1    linear of order 2
ρ711111111-1-111-11-1-11-1-1-1111-11-1    linear of order 2
ρ81111111111-1111111-11-1-1-11-1-1-1    linear of order 2
ρ911111111111-1-111111-111-1-1-1-1-1    linear of order 2
ρ1011111111-1-1-1-111-1-11111-11-1-11-1    linear of order 2
ρ1111111111-1-1-1-1-1-111-111-11111-1-1    linear of order 2
ρ1211111111111-11-1-1-1-11-1-1-1-1111-1    linear of order 2
ρ131111111111-1-11-1-1-1-1-1-11111-1-11    linear of order 2
ρ1411111111-1-11-1-1-111-1-111-1-11-111    linear of order 2
ρ1511111111-1-11-111-1-11-11-11-1-11-11    linear of order 2
ρ161111111111-1-1-11111-1-1-1-11-1111    linear of order 2
ρ1722-2-22-22-2-2200022-2-2000000000    orthogonal lifted from D4
ρ1822-2-22-22-22-20002-22-2000000000    orthogonal lifted from D4
ρ1922-2-22-22-22-2000-22-22000000000    orthogonal lifted from D4
ρ2022-2-22-22-2-22000-2-222000000000    orthogonal lifted from D4
ρ214-4-44-444-4000000000000000000    orthogonal lifted from 2+ (1+4)
ρ224-44-4-4-444000000000000000000    orthogonal lifted from 2+ (1+4)
ρ234-44-444-4-4000000000000000000    orthogonal lifted from 2+ (1+4)
ρ244-4-444-4-44000000000000000000    orthogonal lifted from 2+ (1+4)
ρ254444-4-4-4-4000000000000000000    orthogonal lifted from 2+ (1+4)
ρ2644-4-4-44-44000000000000000000    symplectic lifted from 2- (1+4), Schur index 2

In GAP, Magma, Sage, TeX 

C_2^3._{713}C_2^4
 
% in TeX
 
G:=Group("C2^3.713C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1545);
 
// by ID
 
G=gap.SmallGroup(128,1545);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,758,723,794,185]);
 
// 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=a,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
 

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