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G = C23.602C24order 128 = 27

319th central stem extension by C23 of C24

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

Aliases: C24.63C23, C23.602C24, C22.3762+ 1+4, C22.2802- 1+4, C4⋊C415D4, C23⋊Q847C2, C2.47(Q85D4), C23.79(C4○D4), C2.107(D45D4), C23.7Q891C2, C23.Q864C2, (C23×C4).463C22, (C22×C4).184C23, (C2×C42).654C22, C22.411(C22×D4), C23.23D4.54C2, C23.10D4.44C2, (C22×D4).238C22, (C22×Q8).187C22, C23.81C2389C2, C23.67C2383C2, C24.C22133C2, C23.83C2382C2, C2.68(C22.32C24), C23.65C23123C2, C2.C42.308C22, C2.76(C22.36C24), C2.15(C22.57C24), C2.46(C22.31C24), C2.85(C22.46C24), (C2×C4).104(C2×D4), (C2×C22⋊Q8)⋊42C2, (C2×C422C2)⋊20C2, (C2×C4).428(C4○D4), (C2×C4⋊C4).415C22, C22.464(C2×C4○D4), (C2×C22⋊C4).268C22, SmallGroup(128,1434)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.602C24
C1C2C22C23C22×C4C23×C4C23.7Q8 — C23.602C24
C1C23 — C23.602C24
C1C23 — C23.602C24
C1C23 — C23.602C24

Generators and relations for C23.602C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=ba=ab, e2=b, g2=a, ac=ca, ede-1=ad=da, geg-1=ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, gdg-1=abd, fg=gf >

Subgroups: 500 in 249 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], Q8 [×4], C23, C23 [×2], C23 [×13], C42 [×3], C22⋊C4 [×19], C4⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×13], C22×C4 [×5], C2×D4 [×4], C2×Q8 [×5], C24 [×2], C2.C42 [×10], C2×C42 [×2], C2×C22⋊C4 [×11], C2×C4⋊C4 [×9], C22⋊Q8 [×4], C422C2 [×4], C23×C4, C22×D4, C22×Q8, C23.7Q8, C23.23D4, C24.C22 [×3], C23.65C23, C23.67C23, C23⋊Q8, C23.10D4 [×2], C23.Q8, C23.81C23, C23.83C23, C2×C22⋊Q8, C2×C422C2, C23.602C24
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, C22.32C24, C22.36C24, D45D4, Q85D4, C22.46C24, C22.57C24, C23.602C24

Smallest permutation representation of C23.602C24
On 64 points
Generators in S64
(1 11)(2 12)(3 9)(4 10)(5 62)(6 63)(7 64)(8 61)(13 43)(14 44)(15 41)(16 42)(17 23)(18 24)(19 21)(20 22)(25 31)(26 32)(27 29)(28 30)(33 39)(34 40)(35 37)(36 38)(45 51)(46 52)(47 49)(48 50)(53 59)(54 60)(55 57)(56 58)
(1 9)(2 10)(3 11)(4 12)(5 64)(6 61)(7 62)(8 63)(13 41)(14 42)(15 43)(16 44)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(45 49)(46 50)(47 51)(48 52)(53 57)(54 58)(55 59)(56 60)
(1 59)(2 60)(3 57)(4 58)(5 45)(6 46)(7 47)(8 48)(9 55)(10 56)(11 53)(12 54)(13 30)(14 31)(15 32)(16 29)(17 34)(18 35)(19 36)(20 33)(21 38)(22 39)(23 40)(24 37)(25 44)(26 41)(27 42)(28 43)(49 64)(50 61)(51 62)(52 63)
(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 10 9 2)(3 12 11 4)(5 52 64 48)(6 47 61 51)(7 50 62 46)(8 45 63 49)(13 14 41 42)(15 16 43 44)(17 39 21 35)(18 34 22 38)(19 37 23 33)(20 36 24 40)(25 32 29 28)(26 27 30 31)(53 58 57 54)(55 60 59 56)
(1 63)(2 5)(3 61)(4 7)(6 11)(8 9)(10 64)(12 62)(13 34)(14 39)(15 36)(16 37)(17 30)(18 27)(19 32)(20 25)(21 26)(22 31)(23 28)(24 29)(33 44)(35 42)(38 41)(40 43)(45 60)(46 53)(47 58)(48 55)(49 56)(50 57)(51 54)(52 59)
(1 43 11 13)(2 42 12 16)(3 41 9 15)(4 44 10 14)(5 35 62 37)(6 34 63 40)(7 33 64 39)(8 36 61 38)(17 52 23 46)(18 51 24 45)(19 50 21 48)(20 49 22 47)(25 56 31 58)(26 55 32 57)(27 54 29 60)(28 53 30 59)

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

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

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

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
kernelC23.602C24C23.7Q8C23.23D4C24.C22C23.65C23C23.67C23C23⋊Q8C23.10D4C23.Q8C23.81C23C23.83C23C2×C22⋊Q8C2×C422C2C4⋊C4C2×C4C23C22C22
# reps111311121111144422

Matrix representation of C23.602C24 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
220000
030000
004000
000400
000030
000002
,
330000
420000
004000
004100
000020
000002
,
400000
040000
001300
000400
000001
000010
,
300000
420000
004000
000400
000001
000010

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

C23.602C24 in GAP, Magma, Sage, TeX

C_2^3._{602}C_2^4
% in TeX

G:=Group("C2^3.602C2^4");
// GroupNames label

G:=SmallGroup(128,1434);
// by ID

G=gap.SmallGroup(128,1434);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,232,758,723,1571,346,80]);
// Polycyclic

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

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