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G = C23.426C24order 128 = 27

143rd central stem extension by C23 of C24

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

Aliases: C23.426C24, C24.314C23, C22.2182+ 1+4, C424C420C2, (C22×C4).88C23, (C2×C42).55C22, C23.Q825C2, C23.148(C4○D4), C23.11D438C2, C23.34D434C2, (C23×C4).109C22, C24.C2274C2, C23.23D4.32C2, C23.10D4.16C2, (C22×D4).158C22, C23.63C2378C2, C23.83C2332C2, C23.65C2381C2, C24.3C22.41C2, C2.40(C22.45C24), C2.C42.542C22, C2.20(C22.49C24), C2.69(C23.36C23), C2.22(C22.34C24), C2.48(C22.47C24), (C4×C22⋊C4)⋊81C2, (C2×C4).522(C4○D4), (C2×C4⋊C4).288C22, C22.303(C2×C4○D4), (C2×C22⋊C4).51C22, SmallGroup(128,1258)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.426C24
C1C2C22C23C22×C4C23×C4C4×C22⋊C4 — C23.426C24
C1C23 — C23.426C24
C1C23 — C23.426C24
C1C23 — C23.426C24

Generators and relations for C23.426C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=cb=bc, g2=c, ab=ba, eae-1=ac=ca, faf-1=ad=da, ag=ga, bd=db, fef-1=geg-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 452 in 227 conjugacy classes, 92 normal (82 characteristic)
C1, C2 [×7], C2 [×3], C4 [×17], C22 [×7], C22 [×17], C2×C4 [×8], C2×C4 [×39], D4 [×4], C23, C23 [×2], C23 [×13], C42 [×7], C22⋊C4 [×17], C4⋊C4 [×8], C22×C4 [×13], C22×C4 [×5], C2×D4 [×5], C24 [×2], C2.C42 [×12], C2×C42 [×4], C2×C22⋊C4 [×11], C2×C4⋊C4 [×6], C23×C4, C22×D4, C424C4, C4×C22⋊C4, C23.34D4, C23.23D4, C23.63C23, C24.C22 [×4], C23.65C23, C24.3C22, C23.10D4, C23.Q8, C23.11D4, C23.83C23, C23.426C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×10], C24, C2×C4○D4 [×5], 2+ 1+4 [×2], C23.36C23 [×2], C22.34C24, C22.45C24 [×2], C22.47C24, C22.49C24, C23.426C24

Smallest permutation representation of C23.426C24
On 64 points
Generators in S64
(1 47)(2 39)(3 45)(4 37)(5 11)(6 23)(7 9)(8 21)(10 30)(12 32)(13 49)(14 19)(15 51)(16 17)(18 41)(20 43)(22 31)(24 29)(25 34)(26 63)(27 36)(28 61)(33 55)(35 53)(38 57)(40 59)(42 50)(44 52)(46 60)(48 58)(54 64)(56 62)
(1 23)(2 24)(3 21)(4 22)(5 46)(6 47)(7 48)(8 45)(9 58)(10 59)(11 60)(12 57)(13 54)(14 55)(15 56)(16 53)(17 35)(18 36)(19 33)(20 34)(25 43)(26 44)(27 41)(28 42)(29 39)(30 40)(31 37)(32 38)(49 64)(50 61)(51 62)(52 63)
(1 57)(2 58)(3 59)(4 60)(5 31)(6 32)(7 29)(8 30)(9 24)(10 21)(11 22)(12 23)(13 41)(14 42)(15 43)(16 44)(17 52)(18 49)(19 50)(20 51)(25 56)(26 53)(27 54)(28 55)(33 61)(34 62)(35 63)(36 64)(37 46)(38 47)(39 48)(40 45)
(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 5 12 37)(2 47 9 32)(3 7 10 39)(4 45 11 30)(6 58 38 24)(8 60 40 22)(13 61 27 19)(14 51 28 34)(15 63 25 17)(16 49 26 36)(18 53 64 44)(20 55 62 42)(21 48 59 29)(23 46 57 31)(33 54 50 41)(35 56 52 43)
(1 50 57 19)(2 62 58 34)(3 52 59 17)(4 64 60 36)(5 41 31 13)(6 28 32 55)(7 43 29 15)(8 26 30 53)(9 20 24 51)(10 35 21 63)(11 18 22 49)(12 33 23 61)(14 47 42 38)(16 45 44 40)(25 39 56 48)(27 37 54 46)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J4A···4H4I···4X4Y4Z4AA
order12···22224···44···4444
size11···14482···24···4888

38 irreducible representations

dim1111111111111224
type++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C4○D4C4○D42+ 1+4
kernelC23.426C24C424C4C4×C22⋊C4C23.34D4C23.23D4C23.63C23C24.C22C23.65C23C24.3C22C23.10D4C23.Q8C23.11D4C23.83C23C2×C4C23C22
# reps11111141111111642

Matrix representation of C23.426C24 in GL6(𝔽5)

230000
430000
003100
002200
000042
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
300000
030000
001000
004400
000021
000023
,
140000
040000
002400
003300
000030
000003
,
100000
010000
001200
004400
000030
000003

G:=sub<GL(6,GF(5))| [2,4,0,0,0,0,3,3,0,0,0,0,0,0,3,2,0,0,0,0,1,2,0,0,0,0,0,0,4,0,0,0,0,0,2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,4,0,0,0,0,0,4,0,0,0,0,0,0,2,2,0,0,0,0,1,3],[1,0,0,0,0,0,4,4,0,0,0,0,0,0,2,3,0,0,0,0,4,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,4,0,0,0,0,2,4,0,0,0,0,0,0,3,0,0,0,0,0,0,3] >;

C23.426C24 in GAP, Magma, Sage, TeX

C_2^3._{426}C_2^4
% in TeX

G:=Group("C2^3.426C2^4");
// GroupNames label

G:=SmallGroup(128,1258);
// by ID

G=gap.SmallGroup(128,1258);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,344,758,723,675,136]);
// Polycyclic

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

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