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G = C23.455C24order 128 = 27

172nd central stem extension by C23 of C24

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

Aliases: C23.455C24, C24.328C23, C22.2402+ 1+4, C2.31D42, C22⋊C428D4, C23.53(C2×D4), C232D421C2, C2.75(D45D4), C23.10D444C2, C23.23D457C2, (C23×C4).399C22, (C2×C42).560C22, C22.306(C22×D4), C24.C2282C2, C24.3C2258C2, (C22×C4).1259C23, (C22×D4).170C22, (C22×Q8).135C22, C23.78C2316C2, C2.23(C22.32C24), C2.C42.192C22, C2.35(C22.26C24), C2.25(C22.49C24), (C2×C4⋊D4)⋊20C2, (C4×C22⋊C4)⋊86C2, (C2×C4).907(C2×D4), (C2×C4.4D4)⋊17C2, (C2×C4).388(C4○D4), (C2×C4⋊C4).307C22, C22.331(C2×C4○D4), (C2×C22⋊C4).506C22, SmallGroup(128,1287)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.455C24
C1C2C22C23C22×C4C2×C42C4×C22⋊C4 — C23.455C24
C1C23 — C23.455C24
C1C23 — C23.455C24
C1C23 — C23.455C24

Generators and relations for C23.455C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=ca=ac, f2=a, g2=b, ab=ba, ede-1=gdg-1=ad=da, 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, eg=ge, fg=gf >

Subgroups: 756 in 348 conjugacy classes, 108 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×18], C22 [×3], C22 [×4], C22 [×34], C2×C4 [×14], C2×C4 [×34], D4 [×24], Q8 [×4], C23, C23 [×4], C23 [×26], C42 [×5], C22⋊C4 [×8], C22⋊C4 [×24], C4⋊C4 [×5], C22×C4 [×3], C22×C4 [×8], C22×C4 [×10], C2×D4 [×29], C2×Q8 [×5], C24 [×4], C2.C42 [×2], C2.C42 [×4], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×14], C2×C4⋊C4, C2×C4⋊C4 [×2], C4⋊D4 [×8], C4.4D4 [×8], C23×C4 [×2], C22×D4 [×2], C22×D4 [×4], C22×Q8, C4×C22⋊C4 [×2], C23.23D4 [×2], C24.C22 [×2], C24.3C22, C232D4 [×2], C23.10D4, C23.78C23, C2×C4⋊D4 [×2], C2×C4.4D4 [×2], C23.455C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×6], C24, C22×D4 [×2], C2×C4○D4 [×3], 2+ 1+4 [×2], C22.26C24 [×2], C22.32C24, D42, D45D4 [×2], C22.49C24, C23.455C24

Smallest permutation representation of C23.455C24
On 64 points
Generators in S64
(1 10)(2 11)(3 12)(4 9)(5 37)(6 38)(7 39)(8 40)(13 52)(14 49)(15 50)(16 51)(17 46)(18 47)(19 48)(20 45)(21 43)(22 44)(23 41)(24 42)(25 54)(26 55)(27 56)(28 53)(29 60)(30 57)(31 58)(32 59)(33 64)(34 61)(35 62)(36 63)
(1 26)(2 27)(3 28)(4 25)(5 23)(6 24)(7 21)(8 22)(9 54)(10 55)(11 56)(12 53)(13 60)(14 57)(15 58)(16 59)(17 62)(18 63)(19 64)(20 61)(29 52)(30 49)(31 50)(32 51)(33 48)(34 45)(35 46)(36 47)(37 41)(38 42)(39 43)(40 44)
(1 12)(2 9)(3 10)(4 11)(5 39)(6 40)(7 37)(8 38)(13 50)(14 51)(15 52)(16 49)(17 48)(18 45)(19 46)(20 47)(21 41)(22 42)(23 43)(24 44)(25 56)(26 53)(27 54)(28 55)(29 58)(30 59)(31 60)(32 57)(33 62)(34 63)(35 64)(36 61)
(1 41)(2 24)(3 43)(4 22)(5 55)(6 27)(7 53)(8 25)(9 44)(10 23)(11 42)(12 21)(13 34)(14 62)(15 36)(16 64)(17 57)(18 31)(19 59)(20 29)(26 37)(28 39)(30 46)(32 48)(33 51)(35 49)(38 56)(40 54)(45 60)(47 58)(50 63)(52 61)
(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 18 10 47)(2 46 11 17)(3 20 12 45)(4 48 9 19)(5 58 37 31)(6 30 38 57)(7 60 39 29)(8 32 40 59)(13 43 52 21)(14 24 49 42)(15 41 50 23)(16 22 51 44)(25 33 54 64)(26 63 55 36)(27 35 56 62)(28 61 53 34)
(1 31 26 50)(2 32 27 51)(3 29 28 52)(4 30 25 49)(5 63 23 18)(6 64 24 19)(7 61 21 20)(8 62 22 17)(9 57 54 14)(10 58 55 15)(11 59 56 16)(12 60 53 13)(33 42 48 38)(34 43 45 39)(35 44 46 40)(36 41 47 37)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A···4H4I···4V4W4X
order12···22222224···44···444
size11···14444882···24···488

38 irreducible representations

dim1111111111224
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4C4○D42+ 1+4
kernelC23.455C24C4×C22⋊C4C23.23D4C24.C22C24.3C22C232D4C23.10D4C23.78C23C2×C4⋊D4C2×C4.4D4C22⋊C4C2×C4C22
# reps12221211228122

Matrix representation of C23.455C24 in GL6(𝔽5)

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

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

C23.455C24 in GAP, Magma, Sage, TeX

C_2^3._{455}C_2^4
% in TeX

G:=Group("C2^3.455C2^4");
// GroupNames label

G:=SmallGroup(128,1287);
// by ID

G=gap.SmallGroup(128,1287);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,568,758,723,100,675,136]);
// Polycyclic

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

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