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G = C23.652C24order 128 = 27

369th central stem extension by C23 of C24

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

Aliases: C24.80C23, C23.652C24, C22.4252+ 1+4, C232D4.29C2, C23.4Q860C2, C23.189(C4○D4), C23.34D458C2, (C23×C4).163C22, (C22×C4).573C23, (C2×C42).691C22, C23.10D4100C2, C23.23D4102C2, C23.11D4111C2, C24.3C2292C2, (C22×D4).269C22, C24.C22160C2, C2.85(C22.32C24), C23.63C23167C2, C2.27(C22.54C24), C2.C42.356C22, C2.104(C22.45C24), C2.38(C22.53C24), C2.54(C22.34C24), (C2×C4).216(C4○D4), (C2×C4⋊C4).463C22, C22.513(C2×C4○D4), (C2×C22⋊C4).67C22, SmallGroup(128,1484)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.652C24
C1C2C22C23C24C23×C4C23.23D4 — C23.652C24
C1C23 — C23.652C24
C1C23 — C23.652C24
C1C23 — C23.652C24

Generators and relations for C23.652C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=ca=ac, e2=a, f2=ba=ab, 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: 548 in 242 conjugacy classes, 88 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×14], C22 [×3], C22 [×4], C22 [×24], C2×C4 [×4], C2×C4 [×38], D4 [×12], C23, C23 [×2], C23 [×20], C42 [×2], C22⋊C4 [×16], C4⋊C4 [×7], C22×C4 [×2], C22×C4 [×10], C22×C4 [×4], C2×D4 [×13], C24, C24 [×2], C2.C42 [×2], C2.C42 [×8], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×12], C2×C4⋊C4, C2×C4⋊C4 [×4], C23×C4, C22×D4, C22×D4 [×2], C23.34D4, C23.23D4 [×2], C23.63C23 [×2], C24.C22 [×2], C24.3C22 [×2], C232D4, C23.10D4 [×2], C23.11D4 [×2], C23.4Q8, C23.652C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×6], C24, C2×C4○D4 [×3], 2+ 1+4 [×4], C22.32C24, C22.34C24 [×2], C22.45C24 [×2], C22.53C24, C22.54C24, C23.652C24

Smallest permutation representation of C23.652C24
On 64 points
Generators in S64
(1 36)(2 33)(3 34)(4 35)(5 63)(6 64)(7 61)(8 62)(9 14)(10 15)(11 16)(12 13)(17 22)(18 23)(19 24)(20 21)(25 30)(26 31)(27 32)(28 29)(37 41)(38 42)(39 43)(40 44)(45 60)(46 57)(47 58)(48 59)(49 54)(50 55)(51 56)(52 53)
(1 7)(2 8)(3 5)(4 6)(9 29)(10 30)(11 31)(12 32)(13 27)(14 28)(15 25)(16 26)(17 55)(18 56)(19 53)(20 54)(21 49)(22 50)(23 51)(24 52)(33 62)(34 63)(35 64)(36 61)(37 48)(38 45)(39 46)(40 47)(41 59)(42 60)(43 57)(44 58)
(1 34)(2 35)(3 36)(4 33)(5 61)(6 62)(7 63)(8 64)(9 16)(10 13)(11 14)(12 15)(17 24)(18 21)(19 22)(20 23)(25 32)(26 29)(27 30)(28 31)(37 43)(38 44)(39 41)(40 42)(45 58)(46 59)(47 60)(48 57)(49 56)(50 53)(51 54)(52 55)
(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 39 36 43)(2 44 33 40)(3 37 34 41)(4 42 35 38)(5 48 63 59)(6 60 64 45)(7 46 61 57)(8 58 62 47)(9 24 14 19)(10 20 15 21)(11 22 16 17)(12 18 13 23)(25 49 30 54)(26 55 31 50)(27 51 32 56)(28 53 29 52)
(1 47 61 44)(2 37 62 59)(3 45 63 42)(4 39 64 57)(5 38 34 60)(6 46 35 43)(7 40 36 58)(8 48 33 41)(9 23 28 56)(10 52 25 19)(11 21 26 54)(12 50 27 17)(13 55 32 22)(14 18 29 51)(15 53 30 24)(16 20 31 49)
(1 50)(2 18)(3 52)(4 20)(5 24)(6 54)(7 22)(8 56)(9 48)(10 42)(11 46)(12 44)(13 40)(14 59)(15 38)(16 57)(17 61)(19 63)(21 35)(23 33)(25 45)(26 43)(27 47)(28 41)(29 37)(30 60)(31 39)(32 58)(34 53)(36 55)(49 64)(51 62)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4P4Q4R4S4T
order12···222224···44444
size11···144884···48888

32 irreducible representations

dim1111111111224
type+++++++++++
imageC1C2C2C2C2C2C2C2C2C2C4○D4C4○D42+ 1+4
kernelC23.652C24C23.34D4C23.23D4C23.63C23C24.C22C24.3C22C232D4C23.10D4C23.11D4C23.4Q8C2×C4C23C22
# reps1122221221844

Matrix representation of C23.652C24 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
210000
030000
003000
000300
000002
000030
,
420000
410000
003400
003200
000010
000001
,
200000
020000
001300
000400
000002
000020
,
400000
410000
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],[2,0,0,0,0,0,1,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,2,0],[4,4,0,0,0,0,2,1,0,0,0,0,0,0,3,3,0,0,0,0,4,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,3,4,0,0,0,0,0,0,0,2,0,0,0,0,2,0],[4,4,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C23.652C24 in GAP, Magma, Sage, TeX

C_2^3._{652}C_2^4
% in TeX

G:=Group("C2^3.652C2^4");
// GroupNames label

G:=SmallGroup(128,1484);
// by ID

G=gap.SmallGroup(128,1484);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,120,758,723,268,1571,346]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=c*a=a*c,e^2=a,f^2=b*a=a*b,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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