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G = C23.608C24order 128 = 27

325th central stem extension by C23 of C24

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

Aliases: C23.608C24, C24.410C23, C22.3822+ 1+4, C22.2852- 1+4, (C2×D4).142D4, C23.69(C2×D4), C2.65(D46D4), C23.81(C4○D4), C2.113(D45D4), C23.7Q896C2, C23.23D495C2, C23.11D492C2, C23.10D491C2, C2.48(C233D4), (C22×C4).186C23, (C23×C4).467C22, C23.8Q8111C2, C22.417(C22×D4), (C22×D4).243C22, C23.81C2393C2, C23.83C2384C2, C2.70(C22.32C24), C2.C42.314C22, C2.70(C22.33C24), C2.20(C22.56C24), C2.86(C22.47C24), (C2×C4).422(C2×D4), (C2×C4⋊D4).47C2, (C2×C4).432(C4○D4), (C2×C4⋊C4).421C22, C22.470(C2×C4○D4), (C2×C22.D4)⋊41C2, (C2×C22⋊C4).274C22, SmallGroup(128,1440)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.608C24
C1C2C22C23C24C23×C4C23.23D4 — C23.608C24
C1C23 — C23.608C24
C1C23 — C23.608C24
C1C23 — C23.608C24

Generators and relations for C23.608C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=g2=1, f2=ba=ab, ac=ca, ede=ad=da, geg=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=abd, fg=gf >

Subgroups: 612 in 284 conjugacy classes, 96 normal (82 characteristic)
C1, C2 [×7], C2 [×6], C4 [×14], C22 [×7], C22 [×30], C2×C4 [×4], C2×C4 [×46], D4 [×12], C23, C23 [×6], C23 [×18], C22⋊C4 [×20], C4⋊C4 [×10], C22×C4 [×12], C22×C4 [×12], C2×D4 [×4], C2×D4 [×12], C24 [×3], C2.C42 [×10], C2×C22⋊C4 [×12], C2×C4⋊C4 [×7], C4⋊D4 [×4], C22.D4 [×4], C23×C4 [×3], C22×D4 [×3], C23.7Q8, C23.8Q8 [×3], C23.23D4 [×3], C23.10D4 [×2], C23.11D4 [×2], C23.81C23, C23.83C23, C2×C4⋊D4, C2×C22.D4, C23.608C24
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 [×3], 2- 1+4, C233D4, C22.32C24, C22.33C24, D45D4, D46D4, C22.47C24, C22.56C24, C23.608C24

Smallest permutation representation of C23.608C24
On 64 points
Generators in S64
(1 25)(2 26)(3 27)(4 28)(5 62)(6 63)(7 64)(8 61)(9 15)(10 16)(11 13)(12 14)(17 23)(18 24)(19 21)(20 22)(29 43)(30 44)(31 41)(32 42)(33 39)(34 40)(35 37)(36 38)(45 57)(46 58)(47 59)(48 60)(49 55)(50 56)(51 53)(52 54)
(1 27)(2 28)(3 25)(4 26)(5 64)(6 61)(7 62)(8 63)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(29 41)(30 42)(31 43)(32 44)(33 37)(34 38)(35 39)(36 40)(45 59)(46 60)(47 57)(48 58)(49 53)(50 54)(51 55)(52 56)
(1 22)(2 23)(3 24)(4 21)(5 10)(6 11)(7 12)(8 9)(13 63)(14 64)(15 61)(16 62)(17 26)(18 27)(19 28)(20 25)(29 34)(30 35)(31 36)(32 33)(37 44)(38 41)(39 42)(40 43)(45 56)(46 53)(47 54)(48 55)(49 60)(50 57)(51 58)(52 59)
(1 59)(2 46)(3 57)(4 48)(5 44)(6 29)(7 42)(8 31)(9 36)(10 37)(11 34)(12 39)(13 40)(14 33)(15 38)(16 35)(17 51)(18 56)(19 49)(20 54)(21 55)(22 52)(23 53)(24 50)(25 47)(26 58)(27 45)(28 60)(30 62)(32 64)(41 61)(43 63)
(1 37)(2 41)(3 39)(4 43)(5 54)(6 48)(7 56)(8 46)(9 53)(10 47)(11 55)(12 45)(13 49)(14 57)(15 51)(16 59)(17 36)(18 32)(19 34)(20 30)(21 40)(22 44)(23 38)(24 42)(25 35)(26 31)(27 33)(28 29)(50 64)(52 62)(58 61)(60 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)
(5 64)(6 61)(7 62)(8 63)(9 13)(10 14)(11 15)(12 16)(29 43)(30 44)(31 41)(32 42)(33 39)(34 40)(35 37)(36 38)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)

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

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

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

32 conjugacy classes

class 1 2A···2G2H···2M4A···4L4M···4R
order12···22···24···44···4
size11···14···44···48···8

32 irreducible representations

dim111111111122244
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+42- 1+4
kernelC23.608C24C23.7Q8C23.8Q8C23.23D4C23.10D4C23.11D4C23.81C23C23.83C23C2×C4⋊D4C2×C22.D4C2×D4C2×C4C23C22C22
# reps113322111144431

Matrix representation of C23.608C24 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
340000
320000
002300
004300
000010
000001
,
420000
010000
001000
000100
000001
000010
,
300000
030000
003000
001200
000040
000001
,
100000
140000
001000
002400
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],[3,3,0,0,0,0,4,2,0,0,0,0,0,0,2,4,0,0,0,0,3,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,1,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,4,0,0,0,0,0,0,1,2,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C23.608C24 in GAP, Magma, Sage, TeX

C_2^3._{608}C_2^4
% in TeX

G:=Group("C2^3.608C2^4");
// GroupNames label

G:=SmallGroup(128,1440);
// by ID

G=gap.SmallGroup(128,1440);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,723,268,1571,346]);
// Polycyclic

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

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