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G = C23.708C24order 128 = 27

425th central stem extension by C23 of C24

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

Aliases: C24.99C23, C23.708C24, C22.3682- 1+4, C22.4812+ 1+4, C23⋊Q859C2, (C22×C4).614C23, (C2×C42).116C22, C23.11D4125C2, C23.10D4.70C2, (C22×D4).290C22, (C22×Q8).228C22, C24.C22177C2, C24.3C22.79C2, C23.63C23196C2, C23.67C23105C2, C23.65C23163C2, C2.112(C22.32C24), C2.C42.412C22, C2.49(C22.49C24), C2.77(C22.50C24), C2.47(C22.56C24), C2.49(C22.53C24), C2.119(C22.33C24), (C2×C4).249(C4○D4), (C2×C4⋊C4).518C22, C22.569(C2×C4○D4), (C2×C22⋊C4).331C22, SmallGroup(128,1540)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.708C24
C1C2C22C23C22×C4C2×C42C24.3C22 — C23.708C24
C1C23 — C23.708C24
C1C23 — C23.708C24
C1C23 — C23.708C24

Generators and relations for C23.708C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=ca=ac, e2=b, f2=cb=bc, g2=ba=ab, ede-1=ad=da, geg-1=ae=ea, af=fa, ag=ga, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, gdg-1=abd, fg=gf >

Subgroups: 452 in 214 conjugacy classes, 88 normal (30 characteristic)
C1, C2 [×7], C2 [×2], C4 [×16], C22 [×7], C22 [×14], C2×C4 [×6], C2×C4 [×36], D4 [×4], Q8 [×4], C23, C23 [×14], C42 [×3], C22⋊C4 [×16], C4⋊C4 [×9], C22×C4 [×3], C22×C4 [×10], C2×D4 [×4], C2×Q8 [×4], C24 [×2], C2.C42 [×12], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×12], C2×C4⋊C4 [×4], C2×C4⋊C4 [×2], C22×D4, C22×Q8, C23.63C23 [×2], C24.C22 [×4], C23.65C23, C24.3C22, C23.67C23, C23⋊Q8 [×2], C23.10D4 [×2], C23.11D4 [×2], C23.708C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×6], C24, C2×C4○D4 [×3], 2+ 1+4 [×3], 2- 1+4, C22.32C24 [×2], C22.33C24, C22.49C24, C22.50C24, C22.53C24, C22.56C24, C23.708C24

Smallest permutation representation of C23.708C24
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 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 47 26 36)(2 19 27 64)(3 45 28 34)(4 17 25 62)(5 50 23 31)(6 16 24 59)(7 52 21 29)(8 14 22 57)(9 46 54 35)(10 18 55 63)(11 48 56 33)(12 20 53 61)(13 43 60 39)(15 41 58 37)(30 40 49 44)(32 38 51 42)
(1 61 53 47)(2 17 54 33)(3 63 55 45)(4 19 56 35)(5 58 43 52)(6 16 44 30)(7 60 41 50)(8 14 42 32)(9 48 27 62)(10 34 28 18)(11 46 25 64)(12 36 26 20)(13 37 31 21)(15 39 29 23)(22 57 38 51)(24 59 40 49)
(1 58 55 50)(2 51 56 59)(3 60 53 52)(4 49 54 57)(5 63 41 47)(6 48 42 64)(7 61 43 45)(8 46 44 62)(9 14 25 30)(10 31 26 15)(11 16 27 32)(12 29 28 13)(17 22 35 40)(18 37 36 23)(19 24 33 38)(20 39 34 21)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I4A···4R4S4T4U4V
order12···2224···44444
size11···1884···48888

32 irreducible representations

dim111111111244
type++++++++++-
imageC1C2C2C2C2C2C2C2C2C4○D42+ 1+42- 1+4
kernelC23.708C24C23.63C23C24.C22C23.65C23C24.3C22C23.67C23C23⋊Q8C23.10D4C23.11D4C2×C4C22C22
# reps1241112221231

Matrix representation of C23.708C24 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
310000
020000
000100
001000
000020
000002
,
400000
110000
003000
000300
000001
000010
,
400000
040000
002000
000300
000001
000040
,
430000
110000
002000
000300
000010
000001

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

C23.708C24 in GAP, Magma, Sage, TeX

C_2^3._{708}C_2^4
% in TeX

G:=Group("C2^3.708C2^4");
// GroupNames label

G:=SmallGroup(128,1540);
// by ID

G=gap.SmallGroup(128,1540);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,232,758,723,520,1571,346,192]);
// Polycyclic

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

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