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G = C23.607C24order 128 = 27

324th central stem extension by C23 of C24

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

Aliases: C24.66C23, C23.607C24, C22.2842- 1+4, C22.3812+ 1+4, C4⋊C4.120D4, C2.48(Q85D4), C2.112(D45D4), C23.7Q895C2, C23.Q867C2, C23.178(C4○D4), C23.11D491C2, (C23×C4).466C22, (C22×C4).882C23, (C2×C42).659C22, C23.8Q8110C2, C22.416(C22×D4), C23.10D4.45C2, (C22×D4).242C22, C24.C22137C2, C23.81C2392C2, C24.3C22.62C2, C23.65C23126C2, C23.63C23138C2, C2.C42.313C22, C2.49(C22.31C24), C2.69(C22.33C24), C2.16(C22.57C24), C2.86(C22.46C24), C2.45(C22.34C24), (C2×C4).109(C2×D4), (C2×C42.C2)⋊23C2, (C2×C4).195(C4○D4), (C2×C4⋊C4).420C22, C22.469(C2×C4○D4), (C2×C22⋊C4).273C22, (C2×C22.D4).27C2, SmallGroup(128,1439)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.607C24
C1C2C22C23C22×C4C23×C4C23.7Q8 — C23.607C24
C1C23 — C23.607C24
C1C23 — C23.607C24
C1C23 — C23.607C24

Generators and relations for C23.607C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=e2=ba=ab, f2=g2=a, ac=ca, ede-1=ad=da, geg-1=ae=ea, 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, gdg-1=abd, fg=gf >

Subgroups: 484 in 245 conjugacy classes, 96 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 [×3], C22⋊C4 [×17], C4⋊C4 [×4], C4⋊C4 [×18], C22×C4 [×13], C22×C4 [×5], C2×D4 [×5], C24 [×2], C2.C42 [×8], C2×C42 [×2], C2×C22⋊C4 [×11], C2×C4⋊C4 [×12], C22.D4 [×4], C42.C2 [×4], C23×C4, C22×D4, C23.7Q8, C23.8Q8, C23.63C23, C24.C22 [×2], C23.65C23, C24.3C22, C23.10D4, C23.Q8 [×3], C23.11D4, C23.81C23, C2×C22.D4, C2×C42.C2, C23.607C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], 2+ 1+4 [×2], 2- 1+4 [×2], C22.31C24, C22.33C24, C22.34C24, D45D4, Q85D4, C22.46C24, C22.57C24, C23.607C24

Smallest permutation representation of C23.607C24
On 64 points
Generators in S64
(1 25)(2 26)(3 27)(4 28)(5 62)(6 63)(7 64)(8 61)(9 14)(10 15)(11 16)(12 13)(17 23)(18 24)(19 21)(20 22)(29 51)(30 52)(31 49)(32 50)(33 39)(34 40)(35 37)(36 38)(41 53)(42 54)(43 55)(44 56)(45 59)(46 60)(47 57)(48 58)
(1 27)(2 28)(3 25)(4 26)(5 64)(6 61)(7 62)(8 63)(9 16)(10 13)(11 14)(12 15)(17 21)(18 22)(19 23)(20 24)(29 49)(30 50)(31 51)(32 52)(33 37)(34 38)(35 39)(36 40)(41 55)(42 56)(43 53)(44 54)(45 57)(46 58)(47 59)(48 60)
(1 51)(2 52)(3 49)(4 50)(5 42)(6 43)(7 44)(8 41)(9 40)(10 37)(11 38)(12 39)(13 33)(14 34)(15 35)(16 36)(17 60)(18 57)(19 58)(20 59)(21 48)(22 45)(23 46)(24 47)(25 29)(26 30)(27 31)(28 32)(53 61)(54 62)(55 63)(56 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 55 3 53)(2 44 4 42)(5 52 7 50)(6 31 8 29)(9 60 11 58)(10 47 12 45)(13 59 15 57)(14 46 16 48)(17 38 19 40)(18 33 20 35)(21 34 23 36)(22 37 24 39)(25 43 27 41)(26 56 28 54)(30 64 32 62)(49 61 51 63)
(1 26 25 2)(3 28 27 4)(5 53 62 41)(6 44 63 56)(7 55 64 43)(8 42 61 54)(9 10 14 15)(11 12 16 13)(17 47 23 57)(18 60 24 46)(19 45 21 59)(20 58 22 48)(29 52 51 30)(31 50 49 32)(33 38 39 36)(34 35 40 37)
(1 15 25 10)(2 14 26 9)(3 13 27 12)(4 16 28 11)(5 19 62 21)(6 18 63 24)(7 17 64 23)(8 20 61 22)(29 37 51 35)(30 40 52 34)(31 39 49 33)(32 38 50 36)(41 59 53 45)(42 58 54 48)(43 57 55 47)(44 60 56 46)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim111111111111122244
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+42- 1+4
kernelC23.607C24C23.7Q8C23.8Q8C23.63C23C24.C22C23.65C23C24.3C22C23.10D4C23.Q8C23.11D4C23.81C23C2×C22.D4C2×C42.C2C4⋊C4C2×C4C23C22C22
# reps111121113111144422

Matrix representation of C23.607C24 in GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
040000
100000
001000
000100
000020
000003
,
200000
020000
004300
000100
000003
000030
,
010000
100000
001000
004400
000030
000003
,
010000
100000
004000
000400
000001
000040

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

C23.607C24 in GAP, Magma, Sage, TeX

C_2^3._{607}C_2^4
% in TeX

G:=Group("C2^3.607C2^4");
// GroupNames label

G:=SmallGroup(128,1439);
// by ID

G=gap.SmallGroup(128,1439);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,344,758,723,100,1571,346,80]);
// Polycyclic

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

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