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G = C23.581C24order 128 = 27

298th central stem extension by C23 of C24

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

Aliases: C24.61C23, C23.581C24, C22.2642- 1+4, C22.3552+ 1+4, C2.45D42, C4⋊C410D4, C22⋊C4.10D4, C23⋊Q840C2, C23.207(C2×D4), C2.88(D45D4), C2.44(Q85D4), C23.75(C4○D4), C23.8Q898C2, C23.Q852C2, C23.23D482C2, C23.10D477C2, (C2×C42).638C22, (C22×C4).868C23, (C23×C4).449C22, C22.390(C22×D4), C24.3C2275C2, (C22×D4).220C22, (C22×Q8).177C22, C24.C22121C2, C23.81C2378C2, C23.65C23115C2, C2.C42.290C22, C2.9(C22.56C24), C2.40(C22.31C24), C2.58(C22.33C24), C2.55(C23.38C23), (C2×C4).89(C2×D4), (C2×C22⋊Q8)⋊36C2, (C2×C4⋊D4).43C2, (C2×C4⋊C4).397C22, C22.444(C2×C4○D4), (C2×C22.D4)⋊32C2, (C2×C22⋊C4).250C22, SmallGroup(128,1413)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.581C24
C1C2C22C23C24C23×C4C2×C22.D4 — C23.581C24
C1C23 — C23.581C24
C1C23 — C23.581C24
C1C23 — C23.581C24

Generators and relations for C23.581C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=b, e2=f2=a, ab=ba, ac=ca, ede-1=ad=da, geg=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=abd, fg=gf >

Subgroups: 644 in 309 conjugacy classes, 104 normal (82 characteristic)
C1, C2 [×7], C2 [×5], C4 [×17], C22 [×7], C22 [×27], C2×C4 [×10], C2×C4 [×39], D4 [×12], Q8 [×4], C23, C23 [×4], C23 [×19], C42, C22⋊C4 [×4], C22⋊C4 [×20], C4⋊C4 [×4], C4⋊C4 [×15], C22×C4 [×12], C22×C4 [×10], C2×D4 [×15], C2×Q8 [×5], C24 [×3], C2.C42 [×6], C2×C42, C2×C22⋊C4 [×13], C2×C4⋊C4 [×9], C4⋊D4 [×4], C22⋊Q8 [×8], C22.D4 [×4], C23×C4 [×2], C22×D4 [×3], C22×Q8, C23.8Q8, C23.23D4, C24.C22, C23.65C23, C24.3C22, C23⋊Q8, C23.10D4 [×3], C23.Q8, C23.81C23, C2×C4⋊D4, C2×C22⋊Q8 [×2], C2×C22.D4, C23.581C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C22×D4 [×2], C2×C4○D4, 2+ 1+4 [×2], 2- 1+4 [×2], C23.38C23, C22.31C24, C22.33C24, D42, D45D4, Q85D4, C22.56C24, C23.581C24

Smallest permutation representation of C23.581C24
On 64 points
Generators in S64
(1 39)(2 40)(3 37)(4 38)(5 17)(6 18)(7 19)(8 20)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 62)(34 63)(35 64)(36 61)
(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 51)(2 52)(3 49)(4 50)(5 36)(6 33)(7 34)(8 35)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)
(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 11 39 43)(2 44 40 12)(3 9 37 41)(4 42 38 10)(5 57 17 29)(6 30 18 58)(7 59 19 31)(8 32 20 60)(13 61 45 36)(14 33 46 62)(15 63 47 34)(16 35 48 64)(21 25 49 53)(22 54 50 26)(23 27 51 55)(24 56 52 28)
(1 47 39 15)(2 46 40 14)(3 45 37 13)(4 48 38 16)(5 41 17 9)(6 44 18 12)(7 43 19 11)(8 42 20 10)(21 57 49 29)(22 60 50 32)(23 59 51 31)(24 58 52 30)(25 61 53 36)(26 64 54 35)(27 63 55 34)(28 62 56 33)
(1 11)(2 42)(3 9)(4 44)(5 45)(6 16)(7 47)(8 14)(10 40)(12 38)(13 17)(15 19)(18 48)(20 46)(21 25)(22 56)(23 27)(24 54)(26 52)(28 50)(29 36)(30 64)(31 34)(32 62)(33 60)(35 58)(37 41)(39 43)(49 53)(51 55)(57 61)(59 63)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K2L4A···4N4O···4S
order12···2222224···44···4
size11···1444484···48···8

32 irreducible representations

dim111111111111122244
type++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D42+ 1+42- 1+4
kernelC23.581C24C23.8Q8C23.23D4C24.C22C23.65C23C24.3C22C23⋊Q8C23.10D4C23.Q8C23.81C23C2×C4⋊D4C2×C22⋊Q8C2×C22.D4C22⋊C4C4⋊C4C23C22C22
# reps111111131112144422

Matrix representation of C23.581C24 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
040000
002200
000300
000010
000001
,
040000
100000
004000
000400
000040
000001
,
200000
020000
004000
002100
000001
000010
,
010000
100000
004000
002100
000040
000004

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,2,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,4,2,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,2,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;

C23.581C24 in GAP, Magma, Sage, TeX

C_2^3._{581}C_2^4
% in TeX

G:=Group("C2^3.581C2^4");
// GroupNames label

G:=SmallGroup(128,1413);
// by ID

G=gap.SmallGroup(128,1413);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,120,758,723,100,1571,346]);
// Polycyclic

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

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