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G = C24.346C23order 128 = 27

186th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.346C23, C23.489C24, C22.2712+ 1+4, C22⋊C421Q8, C23.30(C2×Q8), C2.47(D43Q8), (C2×C42).74C22, C23⋊Q8.11C2, (C23×C4).127C22, (C22×C4).548C23, C23.8Q8.38C2, C23.4Q8.10C2, C22.124(C22×Q8), (C22×Q8).146C22, C23.65C2394C2, C23.83C2348C2, C23.63C2398C2, C23.81C2348C2, C23.67C2367C2, C2.32(C22.32C24), C24.C22.37C2, C2.62(C22.45C24), C2.C42.223C22, C2.35(C23.37C23), C2.24(C22.53C24), C2.93(C23.36C23), (C4×C4⋊C4)⋊104C2, (C2×C4).258(C2×Q8), (C4×C22⋊C4).70C2, (C2×C4).401(C4○D4), (C2×C4⋊C4).333C22, C22.365(C2×C4○D4), (C2×C22⋊C4).513C22, SmallGroup(128,1321)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.346C23
C1C2C22C23C22×C4C2×C42C4×C4⋊C4 — C24.346C23
C1C23 — C24.346C23
C1C23 — C24.346C23
C1C23 — C24.346C23

Generators and relations for C24.346C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=cb=bc, f2=c, g2=ba=ab, ac=ca, ede-1=gdg-1=ad=da, ae=ea, af=fa, ag=ga, 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: 404 in 218 conjugacy classes, 100 normal (82 characteristic)
C1, C2 [×7], C2 [×2], C4 [×20], C22 [×7], C22 [×10], C2×C4 [×12], C2×C4 [×40], Q8 [×4], C23, C23 [×2], C23 [×6], C42 [×5], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×14], C22×C4 [×14], C22×C4 [×4], C2×Q8 [×5], C24, C2.C42 [×14], C2×C42 [×4], C2×C22⋊C4 [×6], C2×C4⋊C4 [×9], C23×C4, C22×Q8, C4×C22⋊C4, C4×C4⋊C4, C23.8Q8 [×2], C23.63C23 [×2], C24.C22 [×2], C23.65C23, C23.67C23 [×2], C23⋊Q8, C23.81C23, C23.4Q8, C23.83C23, C24.346C23
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×8], C24, C22×Q8, C2×C4○D4 [×4], 2+ 1+4 [×2], C23.36C23, C23.37C23, C22.32C24, C22.45C24, D43Q8 [×2], C22.53C24, C24.346C23

Smallest permutation representation of C24.346C23
On 64 points
Generators in S64
(1 23)(2 24)(3 21)(4 22)(5 37)(6 38)(7 39)(8 40)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(36 64)
(1 5)(2 6)(3 7)(4 8)(9 55)(10 56)(11 53)(12 54)(13 59)(14 60)(15 57)(16 58)(17 63)(18 64)(19 61)(20 62)(21 39)(22 40)(23 37)(24 38)(25 43)(26 44)(27 41)(28 42)(29 47)(30 48)(31 45)(32 46)(33 51)(34 52)(35 49)(36 50)
(1 7)(2 8)(3 5)(4 6)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)(33 49)(34 50)(35 51)(36 52)
(2 24)(4 22)(6 38)(8 40)(9 41)(11 43)(13 59)(14 32)(15 57)(16 30)(17 35)(18 64)(19 33)(20 62)(25 53)(27 55)(29 47)(31 45)(34 52)(36 50)(46 60)(48 58)(49 63)(51 61)
(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 15 7 59)(2 60 8 16)(3 13 5 57)(4 58 6 14)(9 61 53 17)(10 18 54 62)(11 63 55 19)(12 20 56 64)(21 45 37 29)(22 30 38 46)(23 47 39 31)(24 32 40 48)(25 49 41 33)(26 34 42 50)(27 51 43 35)(28 36 44 52)
(1 41 37 55)(2 42 38 56)(3 43 39 53)(4 44 40 54)(5 27 23 9)(6 28 24 10)(7 25 21 11)(8 26 22 12)(13 35 31 17)(14 36 32 18)(15 33 29 19)(16 34 30 20)(45 63 59 49)(46 64 60 50)(47 61 57 51)(48 62 58 52)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I4A···4H4I···4X4Y4Z4AA4AB
order12···2224···44···44444
size11···1442···24···48888

38 irreducible representations

dim111111111111224
type++++++++++++-+
imageC1C2C2C2C2C2C2C2C2C2C2C2Q8C4○D42+ 1+4
kernelC24.346C23C4×C22⋊C4C4×C4⋊C4C23.8Q8C23.63C23C24.C22C23.65C23C23.67C23C23⋊Q8C23.81C23C23.4Q8C23.83C23C22⋊C4C2×C4C22
# reps1112221211114162

Matrix representation of C24.346C23 in GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
400000
040000
004000
000400
000010
000001
,
100000
010000
001000
000400
000010
000004
,
030000
300000
001000
000400
000002
000030
,
010000
400000
000100
004000
000040
000004
,
100000
010000
003000
000300
000001
000040

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

C24.346C23 in GAP, Magma, Sage, TeX

C_2^4._{346}C_2^3
% in TeX

G:=Group("C2^4.346C2^3");
// GroupNames label

G:=SmallGroup(128,1321);
// by ID

G=gap.SmallGroup(128,1321);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,680,758,723,352,675,192]);
// Polycyclic

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