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G = C24.438C23order 128 = 27

278th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.438C23, C23.657C24, C22.4302+ 1+4, C22.3242- 1+4, C425C431C2, C23.94(C4○D4), C23.Q881C2, (C2×C42).693C22, (C22×C4).577C23, (C23×C4).165C22, C23.8Q8130C2, C23.11D4114C2, C23.23D4.68C2, C23.10D4.59C2, (C22×D4).272C22, C24.C22163C2, C24.3C22.71C2, C23.63C23169C2, C23.65C23142C2, C23.81C23114C2, C2.29(C22.54C24), C2.C42.361C22, C2.109(C22.45C24), C2.99(C22.47C24), C2.99(C22.36C24), C2.91(C22.33C24), (C2×C4).218(C4○D4), (C2×C4⋊C4).468C22, C22.518(C2×C4○D4), (C2×C22⋊C4).307C22, SmallGroup(128,1489)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.438C23
C1C2C22C23C22×C4C2×C42C24.3C22 — C24.438C23
C1C23 — C24.438C23
C1C23 — C24.438C23
C1C23 — C24.438C23

Generators and relations for C24.438C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=db=bd, f2=cb=bc, g2=b, eae-1=gag-1=ab=ba, ac=ca, faf-1=ad=da, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, geg-1=bce, fg=gf >

Subgroups: 452 in 217 conjugacy classes, 88 normal (82 characteristic)
C1, C2 [×7], C2 [×3], C4 [×15], C22 [×7], C22 [×17], C2×C4 [×4], C2×C4 [×41], D4 [×4], C23, C23 [×2], C23 [×13], C42 [×3], C22⋊C4 [×15], C4⋊C4 [×10], C22×C4 [×13], C22×C4 [×3], C2×D4 [×5], C24 [×2], C2.C42 [×12], C2×C42 [×2], C2×C22⋊C4 [×11], C2×C4⋊C4 [×8], C23×C4, C22×D4, C425C4, C23.8Q8 [×2], C23.23D4, C23.63C23, C24.C22 [×2], C23.65C23, C24.3C22, C23.10D4, C23.Q8, C23.11D4 [×3], C23.81C23, C24.438C23
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×6], C24, C2×C4○D4 [×3], 2+ 1+4 [×3], 2- 1+4, C22.33C24, C22.36C24 [×2], C22.45C24, C22.47C24 [×2], C22.54C24, C24.438C23

Smallest permutation representation of C24.438C23
On 64 points
Generators in S64
(1 13)(2 17)(3 15)(4 19)(5 34)(6 40)(7 36)(8 38)(9 52)(10 41)(11 50)(12 43)(14 46)(16 48)(18 47)(20 45)(21 42)(22 51)(23 44)(24 49)(25 57)(26 55)(27 59)(28 53)(29 60)(30 54)(31 58)(32 56)(33 62)(35 64)(37 61)(39 63)
(1 45)(2 46)(3 47)(4 48)(5 63)(6 64)(7 61)(8 62)(9 23)(10 24)(11 21)(12 22)(13 20)(14 17)(15 18)(16 19)(25 30)(26 31)(27 32)(28 29)(33 38)(34 39)(35 40)(36 37)(41 49)(42 50)(43 51)(44 52)(53 60)(54 57)(55 58)(56 59)
(1 58)(2 59)(3 60)(4 57)(5 52)(6 49)(7 50)(8 51)(9 34)(10 35)(11 36)(12 33)(13 31)(14 32)(15 29)(16 30)(17 27)(18 28)(19 25)(20 26)(21 37)(22 38)(23 39)(24 40)(41 64)(42 61)(43 62)(44 63)(45 55)(46 56)(47 53)(48 54)
(1 47)(2 48)(3 45)(4 46)(5 61)(6 62)(7 63)(8 64)(9 21)(10 22)(11 23)(12 24)(13 18)(14 19)(15 20)(16 17)(25 32)(26 29)(27 30)(28 31)(33 40)(34 37)(35 38)(36 39)(41 51)(42 52)(43 49)(44 50)(53 58)(54 59)(55 60)(56 57)
(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 55 52)(2 41 56 6)(3 61 53 50)(4 43 54 8)(5 58 44 45)(7 60 42 47)(9 18 39 29)(10 25 40 16)(11 20 37 31)(12 27 38 14)(13 36 26 21)(15 34 28 23)(17 22 32 33)(19 24 30 35)(46 49 59 64)(48 51 57 62)
(1 20 45 13)(2 32 46 27)(3 18 47 15)(4 30 48 25)(5 36 63 37)(6 22 64 12)(7 34 61 39)(8 24 62 10)(9 42 23 50)(11 44 21 52)(14 56 17 59)(16 54 19 57)(26 55 31 58)(28 53 29 60)(33 49 38 41)(35 51 40 43)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J4A···4P4Q···4U
order12···22224···44···4
size11···14484···48···8

32 irreducible representations

dim1111111111112244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C4○D4C4○D42+ 1+42- 1+4
kernelC24.438C23C425C4C23.8Q8C23.23D4C23.63C23C24.C22C23.65C23C24.3C22C23.10D4C23.Q8C23.11D4C23.81C23C2×C4C23C22C22
# reps1121121111318431

Matrix representation of C24.438C23 in GL6(𝔽5)

100000
040000
000100
001000
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
300000
030000
002000
000300
000033
000042
,
010000
100000
003000
000300
000020
000013
,
100000
010000
000100
004000
000010
000034

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

C24.438C23 in GAP, Magma, Sage, TeX

C_2^4._{438}C_2^3
% in TeX

G:=Group("C2^4.438C2^3");
// GroupNames label

G:=SmallGroup(128,1489);
// by ID

G=gap.SmallGroup(128,1489);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,758,723,268,1571,346,80]);
// Polycyclic

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

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