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G = C23.632C24order 128 = 27

349th central stem extension by C23 of C24

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

Aliases: C23.632C24, C24.423C23, C22.4052+ 1+4, C22.3072- 1+4, (C2×D4)⋊6Q8, (C2×Q8)⋊14D4, C2.34(D4×Q8), C23.37(C2×Q8), C2.53(D43Q8), C2.35(Q86D4), C23.Q874C2, C23.4Q853C2, C2.56(C233D4), (C23×C4).478C22, (C22×C4).890C23, (C2×C42).683C22, C23.8Q8122C2, C2.15(C232Q8), C22.441(C22×D4), C22.150(C22×Q8), C23.23D4.59C2, (C22×D4).258C22, (C22×Q8).201C22, C23.67C2393C2, C24.3C22.66C2, C23.81C23103C2, C23.65C23138C2, C2.C42.338C22, C2.80(C22.33C24), C2.24(C22.56C24), (C2×C4).74(C2×Q8), (C2×C4).126(C2×D4), (C2×C22⋊Q8)⋊46C2, (C2×C4).212(C4○D4), (C2×C4⋊C4).445C22, C22.494(C2×C4○D4), (C2×C22⋊C4).295C22, SmallGroup(128,1464)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.632C24
C1C2C22C23C22×C4C23×C4C23.8Q8 — C23.632C24
C1C23 — C23.632C24
C1C23 — C23.632C24
C1C23 — C23.632C24

Generators and relations for C23.632C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=e2=g2=ba=ab, ac=ca, ede-1=ad=da, geg-1=ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, gdg-1=abd, fg=gf >

Subgroups: 532 in 264 conjugacy classes, 104 normal (30 characteristic)
C1, C2 [×7], C2 [×4], C4 [×18], C22 [×7], C22 [×20], C2×C4 [×10], C2×C4 [×42], D4 [×4], Q8 [×4], C23, C23 [×4], C23 [×12], C42, C22⋊C4 [×16], C4⋊C4 [×19], C22×C4 [×3], C22×C4 [×10], C22×C4 [×8], C2×D4 [×4], C2×D4 [×2], C2×Q8 [×4], C2×Q8 [×2], C24 [×2], C2.C42 [×8], C2×C42, C2×C22⋊C4 [×10], C2×C4⋊C4 [×4], C2×C4⋊C4 [×8], C22⋊Q8 [×8], C23×C4 [×2], C22×D4, C22×Q8, C23.8Q8 [×2], C23.23D4 [×2], C23.65C23, C24.3C22, C23.67C23, C23.Q8 [×2], C23.81C23 [×2], C23.4Q8 [×2], C2×C22⋊Q8 [×2], C23.632C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4 [×3], 2- 1+4, C233D4, C22.33C24, C232Q8, D4×Q8, Q86D4, D43Q8, C22.56C24, C23.632C24

Smallest permutation representation of C23.632C24
On 64 points
Generators in S64
(1 13)(2 14)(3 15)(4 16)(5 43)(6 44)(7 41)(8 42)(9 34)(10 35)(11 36)(12 33)(17 60)(18 57)(19 58)(20 59)(21 56)(22 53)(23 54)(24 55)(25 40)(26 37)(27 38)(28 39)(29 52)(30 49)(31 50)(32 51)(45 62)(46 63)(47 64)(48 61)
(1 15)(2 16)(3 13)(4 14)(5 41)(6 42)(7 43)(8 44)(9 36)(10 33)(11 34)(12 35)(17 58)(18 59)(19 60)(20 57)(21 54)(22 55)(23 56)(24 53)(25 38)(26 39)(27 40)(28 37)(29 50)(30 51)(31 52)(32 49)(45 64)(46 61)(47 62)(48 63)
(1 20)(2 17)(3 18)(4 19)(5 26)(6 27)(7 28)(8 25)(9 54)(10 55)(11 56)(12 53)(13 59)(14 60)(15 57)(16 58)(21 36)(22 33)(23 34)(24 35)(29 46)(30 47)(31 48)(32 45)(37 43)(38 44)(39 41)(40 42)(49 64)(50 61)(51 62)(52 63)
(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 36 3 34)(2 12 4 10)(5 52 7 50)(6 30 8 32)(9 13 11 15)(14 33 16 35)(17 53 19 55)(18 23 20 21)(22 58 24 60)(25 45 27 47)(26 63 28 61)(29 41 31 43)(37 46 39 48)(38 64 40 62)(42 51 44 49)(54 59 56 57)
(1 6)(2 43)(3 8)(4 41)(5 14)(7 16)(9 62)(10 48)(11 64)(12 46)(13 44)(15 42)(17 37)(18 25)(19 39)(20 27)(21 30)(22 52)(23 32)(24 50)(26 60)(28 58)(29 53)(31 55)(33 63)(34 45)(35 61)(36 47)(38 59)(40 57)(49 56)(51 54)
(1 10 3 12)(2 9 4 11)(5 45 7 47)(6 48 8 46)(13 35 15 33)(14 34 16 36)(17 54 19 56)(18 53 20 55)(21 60 23 58)(22 59 24 57)(25 29 27 31)(26 32 28 30)(37 51 39 49)(38 50 40 52)(41 64 43 62)(42 63 44 61)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4N4O···4T
order12···222224···44···4
size11···144444···48···8

32 irreducible representations

dim111111111122244
type++++++++++-++-
imageC1C2C2C2C2C2C2C2C2C2Q8D4C4○D42+ 1+42- 1+4
kernelC23.632C24C23.8Q8C23.23D4C23.65C23C24.3C22C23.67C23C23.Q8C23.81C23C23.4Q8C2×C22⋊Q8C2×D4C2×Q8C2×C4C22C22
# reps122111222244431

Matrix representation of C23.632C24 in GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
400000
040000
001000
000100
000010
000001
,
400000
040000
002400
000300
000002
000020
,
100000
040000
002000
000200
000020
000003
,
010000
100000
002400
003300
000010
000001
,
400000
040000
004300
001100
000001
000040

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

C23.632C24 in GAP, Magma, Sage, TeX

C_2^3._{632}C_2^4
% in TeX

G:=Group("C2^3.632C2^4");
// GroupNames label

G:=SmallGroup(128,1464);
// by ID

G=gap.SmallGroup(128,1464);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,344,758,723,1571,346,192]);
// Polycyclic

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