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G = C42.260C23order 128 = 27

121st non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.260C23, (C4xD4).23C4, (C4xQ8).22C4, C4.62(C8oD4), C4:C8.354C22, (C4xM4(2)):31C2, (C2xC4).641C24, (C4xC8).325C22, (C2xC8).474C23, C42.204(C2xC4), C42.12C4:46C2, C8:C4.152C22, C4.16(C42:C2), C22:C8.228C22, C23.100(C22xC4), (C22xC4).912C23, (C22xC8).508C22, C22.169(C23xC4), C4o2(C42.6C22), C4o2(C42.7C22), C42.7C22:33C2, C42.6C22:36C2, (C2xC42).1107C22, C22.4(C42:C2), C42:C2.290C22, (C2xM4(2)).343C22, C42o(C42.7C22), C42o(C42.6C22), (C2xC4xC8):17C2, C2.11(C2xC8oD4), C4:C4.217(C2xC4), (C4xC4oD4).11C2, C4.292(C2xC4oD4), (C2xD4).227(C2xC4), C22:C4.68(C2xC4), (C2xQ8).205(C2xC4), C4o2((C22xC8):C2), (C2xC4).679(C4oD4), (C22xC4).335(C2xC4), (C2xC4).257(C22xC4), C2.41(C2xC42:C2), (C22xC8):C2.20C2, C42o((C22xC8):C2), (C2xC4oD4).280C22, SmallGroup(128,1654)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.260C23
C1C2C4C2xC4C22xC4C2xC42C4xC4oD4 — C42.260C23
C1C22 — C42.260C23
C1C42 — C42.260C23
C1C2C2C2xC4 — C42.260C23

Generators and relations for C42.260C23
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=a-1, ad=da, eae=ab2, bc=cb, bd=db, be=eb, dcd=a2c, ce=ec, ede=b2d >

Subgroups: 268 in 200 conjugacy classes, 136 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C42, C42, C22:C4, C4:C4, C2xC8, C2xC8, M4(2), C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C4xC8, C8:C4, C22:C8, C4:C8, C2xC42, C2xC42, C42:C2, C42:C2, C4xD4, C4xQ8, C22xC8, C2xM4(2), C2xC4oD4, C2xC4xC8, C4xM4(2), (C22xC8):C2, C42.6C22, C42.12C4, C42.7C22, C4xC4oD4, C42.260C23
Quotients: C1, C2, C4, C22, C2xC4, C23, C22xC4, C4oD4, C24, C42:C2, C8oD4, C23xC4, C2xC4oD4, C2xC42:C2, C2xC8oD4, C42.260C23

Smallest permutation representation of C42.260C23
On 64 points
Generators in S64
(1 33 55 18)(2 19 56 34)(3 35 49 20)(4 21 50 36)(5 37 51 22)(6 23 52 38)(7 39 53 24)(8 17 54 40)(9 48 28 62)(10 63 29 41)(11 42 30 64)(12 57 31 43)(13 44 32 58)(14 59 25 45)(15 46 26 60)(16 61 27 47)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 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 10)(2 30)(3 12)(4 32)(5 14)(6 26)(7 16)(8 28)(9 54)(11 56)(13 50)(15 52)(17 62)(18 41)(19 64)(20 43)(21 58)(22 45)(23 60)(24 47)(25 51)(27 53)(29 55)(31 49)(33 63)(34 42)(35 57)(36 44)(37 59)(38 46)(39 61)(40 48)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 61)(34 62)(35 63)(36 64)(37 57)(38 58)(39 59)(40 60)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4L4M···4R4S···4X8A···8P8Q···8X
order122222224···44···44···48···88···8
size111122441···12···24···42···24···4

56 irreducible representations

dim111111111122
type++++++++
imageC1C2C2C2C2C2C2C2C4C4C4oD4C8oD4
kernelC42.260C23C2xC4xC8C4xM4(2)(C22xC8):C2C42.6C22C42.12C4C42.7C22C4xC4oD4C4xD4C4xQ8C2xC4C4
# reps11122441124816

Matrix representation of C42.260C23 in GL4(F17) generated by

01600
1000
0004
00130
,
4000
0400
00130
00013
,
0200
2000
0090
0009
,
0400
13000
00013
0040
,
0100
1000
0001
0010
G:=sub<GL(4,GF(17))| [0,1,0,0,16,0,0,0,0,0,0,13,0,0,4,0],[4,0,0,0,0,4,0,0,0,0,13,0,0,0,0,13],[0,2,0,0,2,0,0,0,0,0,9,0,0,0,0,9],[0,13,0,0,4,0,0,0,0,0,0,4,0,0,13,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C42.260C23 in GAP, Magma, Sage, TeX

C_4^2._{260}C_2^3
% in TeX

G:=Group("C4^2.260C2^3");
// GroupNames label

G:=SmallGroup(128,1654);
// by ID

G=gap.SmallGroup(128,1654);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,100,1018,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=b,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,e*a*e=a*b^2,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=a^2*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations

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