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G = C24.360C23order 128 = 27

200th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.360C23, C23.515C24, C22.2142- 1+4, C22.2932+ 1+4, (C22×C4)⋊32D4, C4.103C22≀C2, C232D423C2, C23⋊Q827C2, C23.188(C2×D4), C23.7Q874C2, (C23×C4).418C22, (C22×C4).853C23, C22.340(C22×D4), (C22×D4).189C22, (C22×Q8).447C22, C2.32(C22.29C24), C2.C42.243C22, C2.21(C22.31C24), (C2×C4⋊D4)⋊21C2, (C2×C4).375(C2×D4), (C22×C4○D4)⋊6C2, C2.27(C2×C22≀C2), (C2×C4⋊C4).353C22, (C2×C22⋊C4).208C22, SmallGroup(128,1347)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.360C23
C1C2C22C23C22×C4C22×D4C2×C4⋊D4 — C24.360C23
C1C23 — C24.360C23
C1C23 — C24.360C23
C1C23 — C24.360C23

Generators and relations for C24.360C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=1, g2=a, ab=ba, ac=ca, ede=ad=da, ae=ea, gfg-1=af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, dg=gd, eg=ge >

Subgroups: 996 in 466 conjugacy classes, 116 normal (9 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×4], C4 [×12], C22, C22 [×6], C22 [×44], C2×C4 [×12], C2×C4 [×48], D4 [×52], Q8 [×8], C23, C23 [×6], C23 [×32], C22⋊C4 [×18], C4⋊C4 [×6], C22×C4, C22×C4 [×21], C22×C4 [×12], C2×D4 [×48], C2×Q8 [×6], C4○D4 [×32], C24 [×5], C2.C42 [×6], C2×C22⋊C4 [×12], C2×C4⋊C4 [×3], C4⋊D4 [×12], C23×C4 [×3], C22×D4, C22×D4 [×9], C22×Q8, C2×C4○D4 [×12], C23.7Q8 [×3], C232D4 [×6], C23⋊Q8 [×2], C2×C4⋊D4 [×3], C22×C4○D4, C24.360C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], 2+ 1+4 [×3], 2- 1+4, C2×C22≀C2, C22.29C24 [×3], C22.31C24 [×3], C24.360C23

Smallest permutation representation of C24.360C23
On 64 points
Generators in S64
(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 19)(2 20)(3 17)(4 18)(5 9)(6 10)(7 11)(8 12)(13 62)(14 63)(15 64)(16 61)(21 25)(22 26)(23 27)(24 28)(29 39)(30 40)(31 37)(32 38)(33 41)(34 42)(35 43)(36 44)(45 55)(46 56)(47 53)(48 54)(49 57)(50 58)(51 59)(52 60)
(1 27)(2 28)(3 25)(4 26)(5 15)(6 16)(7 13)(8 14)(9 64)(10 61)(11 62)(12 63)(17 21)(18 22)(19 23)(20 24)(29 33)(30 34)(31 35)(32 36)(37 43)(38 44)(39 41)(40 42)(45 49)(46 50)(47 51)(48 52)(53 59)(54 60)(55 57)(56 58)
(1 49)(2 50)(3 51)(4 52)(5 44)(6 41)(7 42)(8 43)(9 36)(10 33)(11 34)(12 35)(13 40)(14 37)(15 38)(16 39)(17 59)(18 60)(19 57)(20 58)(21 53)(22 54)(23 55)(24 56)(25 47)(26 48)(27 45)(28 46)(29 61)(30 62)(31 63)(32 64)
(1 33)(2 34)(3 35)(4 36)(5 58)(6 59)(7 60)(8 57)(9 50)(10 51)(11 52)(12 49)(13 54)(14 55)(15 56)(16 53)(17 43)(18 44)(19 41)(20 42)(21 37)(22 38)(23 39)(24 40)(25 31)(26 32)(27 29)(28 30)(45 63)(46 64)(47 61)(48 62)
(2 4)(5 62)(6 61)(7 64)(8 63)(9 13)(10 16)(11 15)(12 14)(18 20)(22 24)(26 28)(29 33)(30 36)(31 35)(32 34)(37 43)(38 42)(39 41)(40 44)(45 55)(46 54)(47 53)(48 56)(49 57)(50 60)(51 59)(52 58)
(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)

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

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

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

32 conjugacy classes

class 1 2A···2G2H···2M2N2O4A4B4C4D4E···4J4K···4P
order12···22···22244444···44···4
size11···14···48822224···48···8

32 irreducible representations

dim111111244
type++++++++-
imageC1C2C2C2C2C2D42+ 1+42- 1+4
kernelC24.360C23C23.7Q8C232D4C23⋊Q8C2×C4⋊D4C22×C4○D4C22×C4C22C22
# reps1362311231

Matrix representation of C24.360C23 in GL8(ℤ)

10000000
01000000
00100000
00010000
0000-1000
00000-100
000000-10
0000000-1
,
10000000
01000000
00-100000
000-10000
00001000
00000100
00000010
00000001
,
-10000000
0-1000000
00-100000
000-10000
00001000
00000100
00000010
00000001
,
10000000
01000000
001-20000
000-10000
00000010
00000001
00001000
00000100
,
01000000
10000000
001-20000
000-10000
0000-1000
00000-100
00000010
00000001
,
10000000
0-1000000
00-100000
00-110000
00001000
0000-1-100
00000010
000000-1-1
,
10000000
01000000
00100000
00010000
00001200
0000-1-100
00000012
000000-1-1

G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-2,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-2,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,2,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,2,-1] >;

C24.360C23 in GAP, Magma, Sage, TeX

C_2^4._{360}C_2^3
% in TeX

G:=Group("C2^4.360C2^3");
// GroupNames label

G:=SmallGroup(128,1347);
// by ID

G=gap.SmallGroup(128,1347);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,723,185,80]);
// Polycyclic

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

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