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G = C23.625C24order 128 = 27

342nd central stem extension by C23 of C24

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

Aliases: C24.72C23, C23.625C24, C22.3982+ 1+4, C22.3002- 1+4, C4⋊C4.127D4, C2.76(D46D4), C2.53(Q85D4), (C22×C4).196C23, (C2×C42).676C22, C22.434(C22×D4), C23.Q8.31C2, C23.11D4.39C2, C23.83C2388C2, C24.C22.55C2, C23.63C23149C2, C23.81C23100C2, C23.65C23135C2, C2.C42.331C22, C2.27(C22.57C24), C2.47(C22.35C24), C2.90(C22.47C24), C2.53(C22.31C24), (C2×C4).120(C2×D4), (C2×C42.C2)⋊27C2, (C2×C4).206(C4○D4), (C2×C4⋊C4).438C22, C22.487(C2×C4○D4), (C2×C422C2).17C2, (C2×C22⋊C4).289C22, SmallGroup(128,1457)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.625C24
C1C2C22C23C22×C4C2×C42C23.65C23 — C23.625C24
C1C23 — C23.625C24
C1C23 — C23.625C24
C1C23 — C23.625C24

Generators and relations for C23.625C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=ba=ab, e2=f2=b, g2=a, ac=ca, ede-1=ad=da, geg-1=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-1=abd, fg=gf >

Subgroups: 388 in 217 conjugacy classes, 96 normal (82 characteristic)
C1, C2 [×7], C2, C4 [×19], C22 [×7], C22 [×7], C2×C4 [×10], C2×C4 [×37], C23, C23 [×7], C42 [×5], C22⋊C4 [×12], C4⋊C4 [×4], C4⋊C4 [×25], C22×C4 [×14], C24, C2.C42 [×10], C2×C42 [×3], C2×C22⋊C4 [×7], C2×C4⋊C4 [×15], C42.C2 [×4], C422C2 [×4], C23.63C23, C24.C22 [×2], C23.65C23 [×4], C23.Q8 [×3], C23.11D4, C23.81C23, C23.83C23, C2×C42.C2, C2×C422C2, C23.625C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], 2+ 1+4, 2- 1+4 [×3], C22.31C24, C22.35C24 [×2], D46D4, Q85D4, C22.47C24, C22.57C24, C23.625C24

Smallest permutation representation of C23.625C24
On 64 points
Generators in S64
(1 41)(2 42)(3 43)(4 44)(5 35)(6 36)(7 33)(8 34)(9 15)(10 16)(11 13)(12 14)(17 52)(18 49)(19 50)(20 51)(21 48)(22 45)(23 46)(24 47)(25 58)(26 59)(27 60)(28 57)(29 54)(30 55)(31 56)(32 53)(37 62)(38 63)(39 64)(40 61)
(1 43)(2 44)(3 41)(4 42)(5 33)(6 34)(7 35)(8 36)(9 13)(10 14)(11 15)(12 16)(17 50)(18 51)(19 52)(20 49)(21 46)(22 47)(23 48)(24 45)(25 60)(26 57)(27 58)(28 59)(29 56)(30 53)(31 54)(32 55)(37 64)(38 61)(39 62)(40 63)
(1 32)(2 29)(3 30)(4 31)(5 49)(6 50)(7 51)(8 52)(9 60)(10 57)(11 58)(12 59)(13 25)(14 26)(15 27)(16 28)(17 34)(18 35)(19 36)(20 33)(21 61)(22 62)(23 63)(24 64)(37 45)(38 46)(39 47)(40 48)(41 53)(42 54)(43 55)(44 56)
(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 44 43 2)(3 42 41 4)(5 52 33 19)(6 18 34 51)(7 50 35 17)(8 20 36 49)(9 16 13 12)(10 11 14 15)(21 62 46 39)(22 38 47 61)(23 64 48 37)(24 40 45 63)(25 59 60 28)(26 27 57 58)(29 32 56 55)(30 54 53 31)
(1 40 43 63)(2 64 44 37)(3 38 41 61)(4 62 42 39)(5 9 33 13)(6 14 34 10)(7 11 35 15)(8 16 36 12)(17 57 50 26)(18 27 51 58)(19 59 52 28)(20 25 49 60)(21 30 46 53)(22 54 47 31)(23 32 48 55)(24 56 45 29)
(1 16 41 10)(2 15 42 9)(3 14 43 12)(4 13 44 11)(5 37 35 62)(6 40 36 61)(7 39 33 64)(8 38 34 63)(17 23 52 46)(18 22 49 45)(19 21 50 48)(20 24 51 47)(25 56 58 31)(26 55 59 30)(27 54 60 29)(28 53 57 32)

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

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

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

32 conjugacy classes

class 1 2A···2G2H4A···4R4S···4W
order12···224···44···4
size11···184···48···8

32 irreducible representations

dim11111111112244
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2D4C4○D42+ 1+42- 1+4
kernelC23.625C24C23.63C23C24.C22C23.65C23C23.Q8C23.11D4C23.81C23C23.83C23C2×C42.C2C2×C422C2C4⋊C4C2×C4C22C22
# reps11243111114813

Matrix representation of C23.625C24 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
040000
100000
004000
000400
000020
000033
,
010000
100000
004000
000100
000030
000003
,
400000
040000
000100
001000
000024
000003
,
200000
030000
001000
000100
000012
000004

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

C23.625C24 in GAP, Magma, Sage, TeX

C_2^3._{625}C_2^4
% in TeX

G:=Group("C2^3.625C2^4");
// GroupNames label

G:=SmallGroup(128,1457);
// by ID

G=gap.SmallGroup(128,1457);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,232,758,723,184,1571,346,80]);
// Polycyclic

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