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G = C23.593C24order 128 = 27

310th central stem extension by C23 of C24

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

Aliases: C23.593C24, C24.400C23, C22.3672+ 1+4, C22.2732- 1+4, C22⋊C415D4, C23.67(C2×D4), C2.98(D45D4), C232D4.21C2, C23.Q858C2, C23.7Q887C2, C23.171(C4○D4), C23.11D482C2, C23.23D487C2, (C23×C4).149C22, (C22×C4).558C23, C23.8Q8105C2, C22.402(C22×D4), (C22×D4).230C22, C23.83C2378C2, C2.14(C22.54C24), C2.77(C22.45C24), C2.C42.300C22, C2.44(C22.31C24), C2.63(C22.33C24), (C2×C4).420(C2×D4), (C2×C4⋊C4).407C22, C22.455(C2×C4○D4), (C2×C22.D4)⋊34C2, (C2×C22⋊C4).260C22, SmallGroup(128,1425)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.593C24
C1C2C22C23C22×C4C23×C4C23.7Q8 — C23.593C24
C1C23 — C23.593C24
C1C23 — C23.593C24
C1C23 — C23.593C24

Generators and relations for C23.593C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=c, f2=g2=b, eae-1=ab=ba, faf-1=ac=ca, ad=da, gag-1=abc, bc=cb, bd=db, geg-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef-1=de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 612 in 284 conjugacy classes, 96 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×14], C22 [×3], C22 [×4], C22 [×30], C2×C4 [×4], C2×C4 [×46], D4 [×12], C23, C23 [×6], C23 [×18], C22⋊C4 [×4], C22⋊C4 [×15], C4⋊C4 [×11], C22×C4 [×2], C22×C4 [×10], C22×C4 [×13], C2×D4 [×15], C24, C24 [×2], C2.C42 [×2], C2.C42 [×8], C2×C22⋊C4 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4, C2×C4⋊C4 [×6], C22.D4 [×8], C23×C4, C23×C4 [×2], C22×D4, C22×D4 [×2], C23.7Q8 [×2], C23.8Q8, C23.23D4 [×4], C232D4, C23.Q8 [×2], C23.11D4, C23.83C23 [×2], C2×C22.D4 [×2], C23.593C24
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 [×3], 2- 1+4, C22.31C24, C22.33C24 [×2], D45D4 [×2], C22.45C24, C22.54C24, C23.593C24

Smallest permutation representation of C23.593C24
On 64 points
Generators in S64
(1 47)(2 35)(3 45)(4 33)(5 53)(6 51)(7 55)(8 49)(9 50)(10 54)(11 52)(12 56)(13 57)(14 42)(15 59)(16 44)(17 38)(18 31)(19 40)(20 29)(21 36)(22 46)(23 34)(24 48)(25 41)(26 58)(27 43)(28 60)(30 62)(32 64)(37 61)(39 63)
(1 23)(2 24)(3 21)(4 22)(5 9)(6 10)(7 11)(8 12)(13 25)(14 26)(15 27)(16 28)(17 62)(18 63)(19 64)(20 61)(29 37)(30 38)(31 39)(32 40)(33 46)(34 47)(35 48)(36 45)(41 57)(42 58)(43 59)(44 60)(49 56)(50 53)(51 54)(52 55)
(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 27)(2 28)(3 25)(4 26)(5 37)(6 38)(7 39)(8 40)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 24)(17 51)(18 52)(19 49)(20 50)(33 58)(34 59)(35 60)(36 57)(41 45)(42 46)(43 47)(44 48)(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 11 23 7)(2 32 24 40)(3 9 21 5)(4 30 22 38)(6 26 10 14)(8 28 12 16)(13 37 25 29)(15 39 27 31)(17 35 62 48)(18 57 63 41)(19 33 64 46)(20 59 61 43)(34 53 47 50)(36 55 45 52)(42 49 58 56)(44 51 60 54)
(1 7 23 11)(2 12 24 8)(3 5 21 9)(4 10 22 6)(13 29 25 37)(14 38 26 30)(15 31 27 39)(16 40 28 32)(17 44 62 60)(18 57 63 41)(19 42 64 58)(20 59 61 43)(33 49 46 56)(34 53 47 50)(35 51 48 54)(36 55 45 52)

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

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

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

32 conjugacy classes

class 1 2A···2G2H···2M4A···4L4M···4R
order12···22···24···44···4
size11···14···44···48···8

32 irreducible representations

dim1111111112244
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2D4C4○D42+ 1+42- 1+4
kernelC23.593C24C23.7Q8C23.8Q8C23.23D4C232D4C23.Q8C23.11D4C23.83C23C2×C22.D4C22⋊C4C23C22C22
# reps1214121224831

Matrix representation of C23.593C24 in GL6(𝔽5)

100000
010000
001000
001400
000011
000004
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
400000
040000
001000
000100
000010
000001
,
010000
100000
002000
000200
000022
000013
,
100000
040000
002100
002300
000020
000002
,
100000
010000
002100
002300
000030
000042

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

C23.593C24 in GAP, Magma, Sage, TeX

C_2^3._{593}C_2^4
% in TeX

G:=Group("C2^3.593C2^4");
// GroupNames label

G:=SmallGroup(128,1425);
// by ID

G=gap.SmallGroup(128,1425);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,232,758,723,100,1571,346,80]);
// Polycyclic

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

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