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G = C23.419C24order 128 = 27

136th central stem extension by C23 of C24

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

Aliases: C24.21C23, C23.419C24, C22.1612- 1+4, C4⋊C4.232D4, C428C438C2, C2.47(D46D4), C2.35(Q85D4), C23.Q8.9C2, C23.4Q8.6C2, (C2×C42).534C22, (C22×C4).529C23, C22.283(C22×D4), C23.63C2373C2, C23.81C2329C2, C23.65C2378C2, C24.C22.24C2, C2.C42.167C22, C2.28(C22.26C24), C2.20(C22.35C24), C2.50(C22.46C24), C2.62(C23.36C23), (C4×C4⋊C4)⋊79C2, (C2×C4).68(C2×D4), (C2×C42.C2)⋊10C2, (C2×C4).378(C4○D4), (C2×C4⋊C4).282C22, (C2×C422C2).8C2, C22.296(C2×C4○D4), (C2×C22⋊C4).165C22, SmallGroup(128,1251)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.419C24
C1C2C22C23C22×C4C2×C42C4×C4⋊C4 — C23.419C24
C1C23 — C23.419C24
C1C23 — C23.419C24
C1C23 — C23.419C24

Generators and relations for C23.419C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=c, e2=f2=a, g2=b, ab=ba, ac=ca, ede-1=gdg-1=ad=da, 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, eg=ge, fg=gf >

Subgroups: 388 in 227 conjugacy classes, 100 normal (82 characteristic)
C1, C2 [×7], C2, C4 [×21], C22 [×7], C22 [×7], C2×C4 [×14], C2×C4 [×35], C23, C23 [×7], C42 [×10], C22⋊C4 [×14], C4⋊C4 [×4], C4⋊C4 [×24], C22×C4 [×14], C24, C2.C42 [×10], C2×C42 [×5], C2×C22⋊C4 [×7], C2×C4⋊C4 [×13], C42.C2 [×4], C422C2 [×4], C4×C4⋊C4 [×2], C428C4, C23.63C23, C24.C22 [×4], C23.65C23, C23.Q8, C23.81C23 [×2], C23.4Q8, C2×C42.C2, C2×C422C2, C23.419C24
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], C23.36C23, C22.26C24, C22.35C24, D46D4, Q85D4, C22.46C24 [×2], C23.419C24

Smallest permutation representation of C23.419C24
On 64 points
Generators in S64
(1 27)(2 28)(3 25)(4 26)(5 54)(6 55)(7 56)(8 53)(9 52)(10 49)(11 50)(12 51)(13 48)(14 45)(15 46)(16 47)(17 42)(18 43)(19 44)(20 41)(21 39)(22 40)(23 37)(24 38)(29 33)(30 34)(31 35)(32 36)(57 64)(58 61)(59 62)(60 63)
(1 61)(2 62)(3 63)(4 64)(5 35)(6 36)(7 33)(8 34)(9 39)(10 40)(11 37)(12 38)(13 17)(14 18)(15 19)(16 20)(21 52)(22 49)(23 50)(24 51)(25 60)(26 57)(27 58)(28 59)(29 56)(30 53)(31 54)(32 55)(41 47)(42 48)(43 45)(44 46)
(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 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 19 27 44)(2 41 28 20)(3 17 25 42)(4 43 26 18)(5 24 54 38)(6 39 55 21)(7 22 56 40)(8 37 53 23)(9 32 52 36)(10 33 49 29)(11 30 50 34)(12 35 51 31)(13 60 48 63)(14 64 45 57)(15 58 46 61)(16 62 47 59)
(1 51 27 12)(2 21 28 39)(3 49 25 10)(4 23 26 37)(5 13 54 48)(6 18 55 43)(7 15 56 46)(8 20 53 41)(9 62 52 59)(11 64 50 57)(14 32 45 36)(16 30 47 34)(17 31 42 35)(19 29 44 33)(22 60 40 63)(24 58 38 61)
(1 34 61 8)(2 31 62 54)(3 36 63 6)(4 29 64 56)(5 28 35 59)(7 26 33 57)(9 13 39 17)(10 45 40 43)(11 15 37 19)(12 47 38 41)(14 22 18 49)(16 24 20 51)(21 42 52 48)(23 44 50 46)(25 32 60 55)(27 30 58 53)

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

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

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

38 conjugacy classes

class 1 2A···2G2H4A···4H4I···4Z4AA4AB4AC
order12···224···44···4444
size11···182···24···4888

38 irreducible representations

dim11111111111224
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2D4C4○D42- 1+4
kernelC23.419C24C4×C4⋊C4C428C4C23.63C23C24.C22C23.65C23C23.Q8C23.81C23C23.4Q8C2×C42.C2C2×C422C2C4⋊C4C2×C4C22
# reps121141121114162

Matrix representation of C23.419C24 in GL6(𝔽5)

100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000040
000004
,
200000
020000
004000
000100
000041
000031
,
030000
200000
000200
002000
000032
000012
,
010000
100000
002000
000200
000040
000031
,
400000
040000
000100
001000
000030
000003

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

C23.419C24 in GAP, Magma, Sage, TeX

C_2^3._{419}C_2^4
% in TeX

G:=Group("C2^3.419C2^4");
// GroupNames label

G:=SmallGroup(128,1251);
// by ID

G=gap.SmallGroup(128,1251);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,723,100,675,136]);
// Polycyclic

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

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