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G = C23.359C24order 128 = 27

76th central stem extension by C23 of C24

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

Aliases: C23.359C24, C24.280C23, C22.1662+ 1+4, C22⋊C440D4, C23⋊Q811C2, C23.171(C2×D4), C232D4.5C2, C2.45(D45D4), C23.29(C4○D4), (C23×C4).85C22, C23.34D426C2, C23.10D428C2, C23.23D443C2, (C22×C4).812C23, (C2×C42).502C22, C22.239(C22×D4), C24.C2244C2, (C22×D4).517C22, (C22×Q8).108C22, C23.63C2341C2, C23.67C2344C2, C2.31(C22.19C24), C2.15(C22.32C24), C2.C42.116C22, C2.9(C22.49C24), C2.11(C22.53C24), C2.32(C23.36C23), (C2×C4×D4)⋊36C2, (C4×C22⋊C4)⋊62C2, (C2×C4).896(C2×D4), (C2×C4.4D4)⋊10C2, (C2×C4).112(C4○D4), (C2×C4⋊C4).240C22, C22.236(C2×C4○D4), (C2×C22⋊C4).136C22, SmallGroup(128,1191)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.359C24
C1C2C22C23C22×C4C2×C42C4×C22⋊C4 — C23.359C24
C1C23 — C23.359C24
C1C23 — C23.359C24
C1C23 — C23.359C24

Generators and relations for C23.359C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=1, f2=b, g2=c, eae=gag-1=ab=ba, faf-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef-1=de=ed, df=fd, dg=gd, eg=ge, fg=gf >

Subgroups: 612 in 295 conjugacy classes, 100 normal (82 characteristic)
C1, C2 [×7], C2 [×5], C4 [×17], C22 [×7], C22 [×27], C2×C4 [×10], C2×C4 [×39], D4 [×16], Q8 [×4], C23, C23 [×4], C23 [×19], C42 [×5], C22⋊C4 [×4], C22⋊C4 [×20], C4⋊C4 [×5], C22×C4 [×12], C22×C4 [×10], C2×D4 [×15], C2×Q8 [×5], C24 [×3], C2.C42 [×10], C2×C42 [×3], C2×C22⋊C4 [×13], C2×C4⋊C4 [×3], C4×D4 [×4], C4.4D4 [×4], C23×C4 [×2], C22×D4 [×3], C22×Q8, C4×C22⋊C4, C23.34D4, C23.23D4 [×2], C23.63C23, C24.C22 [×3], C23.67C23, C232D4, C23⋊Q8, C23.10D4 [×2], C2×C4×D4, C2×C4.4D4, C23.359C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C22×D4, C2×C4○D4 [×4], 2+ 1+4 [×2], C22.19C24, C23.36C23, C22.32C24, D45D4 [×2], C22.49C24, C22.53C24, C23.359C24

Smallest permutation representation of C23.359C24
On 64 points
Generators in S64
(1 28)(2 13)(3 26)(4 15)(5 54)(6 44)(7 56)(8 42)(9 16)(10 25)(11 14)(12 27)(17 58)(18 48)(19 60)(20 46)(21 41)(22 53)(23 43)(24 55)(29 52)(30 38)(31 50)(32 40)(33 51)(34 37)(35 49)(36 39)(45 64)(47 62)(57 61)(59 63)
(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 9)(2 10)(3 11)(4 12)(5 23)(6 24)(7 21)(8 22)(13 25)(14 26)(15 27)(16 28)(17 62)(18 63)(19 64)(20 61)(29 34)(30 35)(31 36)(32 33)(37 52)(38 49)(39 50)(40 51)(41 56)(42 53)(43 54)(44 55)(45 60)(46 57)(47 58)(48 59)
(1 29)(2 30)(3 31)(4 32)(5 59)(6 60)(7 57)(8 58)(9 34)(10 35)(11 36)(12 33)(13 38)(14 39)(15 40)(16 37)(17 42)(18 43)(19 44)(20 41)(21 46)(22 47)(23 48)(24 45)(25 49)(26 50)(27 51)(28 52)(53 62)(54 63)(55 64)(56 61)
(1 61)(2 53)(3 63)(4 55)(5 52)(6 25)(7 50)(8 27)(9 20)(10 42)(11 18)(12 44)(13 24)(14 46)(15 22)(16 48)(17 35)(19 33)(21 39)(23 37)(26 57)(28 59)(29 56)(30 62)(31 54)(32 64)(34 41)(36 43)(38 45)(40 47)(49 60)(51 58)
(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 13 9 25)(2 14 10 26)(3 15 11 27)(4 16 12 28)(5 64 23 19)(6 61 24 20)(7 62 21 17)(8 63 22 18)(29 38 34 49)(30 39 35 50)(31 40 36 51)(32 37 33 52)(41 60 56 45)(42 57 53 46)(43 58 54 47)(44 59 55 48)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L4A···4H4I···4V4W4X4Y
order12···2222224···44···4444
size11···1444482···24···4888

38 irreducible representations

dim1111111111112224
type++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+4
kernelC23.359C24C4×C22⋊C4C23.34D4C23.23D4C23.63C23C24.C22C23.67C23C232D4C23⋊Q8C23.10D4C2×C4×D4C2×C4.4D4C22⋊C4C2×C4C23C22
# reps11121311121141242

Matrix representation of C23.359C24 in GL6(𝔽5)

320000
120000
001100
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
140000
040000
001000
000100
000001
000010
,
300000
030000
002200
001300
000040
000001
,
140000
040000
003000
000300
000010
000001

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

C23.359C24 in GAP, Magma, Sage, TeX

C_2^3._{359}C_2^4
% in TeX

G:=Group("C2^3.359C2^4");
// GroupNames label

G:=SmallGroup(128,1191);
// by ID

G=gap.SmallGroup(128,1191);
# by ID

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

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

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