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G = C23.432C24order 128 = 27

149th central stem extension by C23 of C24

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

Aliases: C24.25C23, C23.432C24, C22.1712- 1+4, C22.2232+ 1+4, C425C417C2, C424C425C2, C23⋊Q8.9C2, (C2×C42).61C22, (C22×C4).533C23, C23.11D4.18C2, (C22×Q8).127C22, C23.81C2334C2, C23.67C2357C2, C23.63C2383C2, C23.83C2336C2, C24.C22.30C2, C2.44(C22.45C24), C2.C42.545C22, C2.58(C22.46C24), C2.23(C22.49C24), C2.34(C22.50C24), C2.43(C22.36C24), C2.75(C23.36C23), (C4×C4⋊C4)⋊84C2, (C2×C4).145(C4○D4), (C2×C4⋊C4).294C22, C22.309(C2×C4○D4), (C2×C22⋊C4).52C22, SmallGroup(128,1264)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.432C24
C1C2C22C23C22×C4C2×C42C4×C4⋊C4 — C23.432C24
C1C23 — C23.432C24
C1C23 — C23.432C24
C1C23 — C23.432C24

Generators and relations for C23.432C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=c, e2=b, f2=a, g2=ba=ab, 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: 372 in 203 conjugacy classes, 92 normal (82 characteristic)
C1, C2 [×7], C2, C4 [×19], C22 [×7], C22 [×7], C2×C4 [×10], C2×C4 [×37], Q8 [×4], C23, C23 [×7], C42 [×10], C22⋊C4 [×12], C4⋊C4 [×9], C22×C4 [×14], C2×Q8 [×5], C24, C2.C42 [×16], C2×C42 [×5], C2×C22⋊C4 [×7], C2×C4⋊C4 [×6], C22×Q8, C424C4, C4×C4⋊C4, C425C4, C23.63C23, C24.C22 [×5], C23.67C23 [×2], C23⋊Q8, C23.11D4, C23.81C23, C23.83C23, C23.432C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×10], C24, C2×C4○D4 [×5], 2+ 1+4, 2- 1+4, C23.36C23 [×2], C22.36C24, C22.45C24, C22.46C24, C22.49C24, C22.50C24, C23.432C24

Smallest permutation representation of C23.432C24
On 64 points
Generators in S64
(1 19)(2 20)(3 17)(4 18)(5 46)(6 47)(7 48)(8 45)(9 58)(10 59)(11 60)(12 57)(13 54)(14 55)(15 56)(16 53)(21 35)(22 36)(23 33)(24 34)(25 39)(26 40)(27 37)(28 38)(29 43)(30 44)(31 41)(32 42)(49 64)(50 61)(51 62)(52 63)
(1 57)(2 58)(3 59)(4 60)(5 37)(6 38)(7 39)(8 40)(9 20)(10 17)(11 18)(12 19)(13 31)(14 32)(15 29)(16 30)(21 52)(22 49)(23 50)(24 51)(25 48)(26 45)(27 46)(28 47)(33 61)(34 62)(35 63)(36 64)(41 54)(42 55)(43 56)(44 53)
(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 46 57 27)(2 6 58 38)(3 48 59 25)(4 8 60 40)(5 12 37 19)(7 10 39 17)(9 28 20 47)(11 26 18 45)(13 61 31 33)(14 51 32 24)(15 63 29 35)(16 49 30 22)(21 56 52 43)(23 54 50 41)(34 55 62 42)(36 53 64 44)
(1 47 19 6)(2 25 20 39)(3 45 17 8)(4 27 18 37)(5 60 46 11)(7 58 48 9)(10 40 59 26)(12 38 57 28)(13 64 54 49)(14 33 55 23)(15 62 56 51)(16 35 53 21)(22 31 36 41)(24 29 34 43)(30 63 44 52)(32 61 42 50)
(1 61 12 23)(2 51 9 34)(3 63 10 21)(4 49 11 36)(5 41 27 13)(6 32 28 55)(7 43 25 15)(8 30 26 53)(14 47 42 38)(16 45 44 40)(17 52 59 35)(18 64 60 22)(19 50 57 33)(20 62 58 24)(29 39 56 48)(31 37 54 46)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim11111111111244
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C4○D42+ 1+42- 1+4
kernelC23.432C24C424C4C4×C4⋊C4C425C4C23.63C23C24.C22C23.67C23C23⋊Q8C23.11D4C23.81C23C23.83C23C2×C4C22C22
# reps111115211112011

Matrix representation of C23.432C24 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000040
000004
,
100000
010000
001000
000100
000040
000004
,
440000
010000
002100
002300
000022
000003
,
220000
130000
003000
000300
000033
000002
,
200000
020000
004200
000100
000010
000034
,
440000
210000
003000
000300
000030
000003

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

C23.432C24 in GAP, Magma, Sage, TeX

C_2^3._{432}C_2^4
% in TeX

G:=Group("C2^3.432C2^4");
// GroupNames label

G:=SmallGroup(128,1264);
// by ID

G=gap.SmallGroup(128,1264);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,723,100,675,192]);
// Polycyclic

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