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G = C42.291C23order 128 = 27

152nd non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.291C23, (C8xD4):41C2, (C8xQ8):30C2, C8o3(C4:D4), C8o2(C8:9D4), C8o2(C8:6D4), C8:9D4:54C2, C8:6D4:52C2, C8o2(C8:4Q8), C8:4Q8:52C2, C8o3(C22:Q8), C8o(C42:2C2), C4.20(C8oD4), C8o2(C4.4D4), C4:D4.36C4, C8o2(C42.C2), C22:Q8.36C4, C8.106(C4oD4), C4:C8.362C22, C8o(C42.6C4), (C2xC4).663C24, C42:2C2.8C4, C42.287(C2xC4), (C4xC8).438C22, (C2xC8).641C23, C4.4D4.28C4, C22.3(C8oD4), C42.C2.28C4, C8o2M4(2):33C2, C42.6C4:59C2, (C4xD4).294C22, C8o2(C22.D4), C23.38(C22xC4), (C4xQ8).279C22, C8:C4.163C22, C22:C8.233C22, C8o(C42.7C22), (C22xC8).516C22, C22.188(C23xC4), C22.D4.16C4, C8o(C23.36C23), (C2xC42).1120C22, C42.7C22:34C2, (C22xC4).1278C23, C42:C2.306C22, (C2xM4(2)).365C22, C23.36C23.35C2, (C2xC4xC8):45C2, (C2xC8)o(C8:4Q8), C2.24(C2xC8oD4), C2.45(C4xC4oD4), C4:C4.164(C2xC4), (C2xC8)o(C4.4D4), C4.314(C2xC4oD4), (C2xC8)o(C42.C2), (C2xC8)o(C42:2C2), (C2xD4).180(C2xC4), C22:C4.41(C2xC4), (C2xC4).77(C22xC4), (C2xQ8).164(C2xC4), (C2xC8)o(C42.6C4), (C22xC4).388(C2xC4), (C2xC8)o(C22.D4), (C2xC8)o(C42.7C22), (C2xC8)o(C23.36C23), SmallGroup(128,1698)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.291C23
C1C2C4C2xC4C42C4xC8C2xC4xC8 — C42.291C23
C1C22 — C42.291C23
C1C2xC8 — C42.291C23
C1C2C2C2xC4 — C42.291C23

Generators and relations for C42.291C23
 G = < a,b,c,d,e | a4=b4=1, c2=b2, d2=a2b-1, e2=a2b2, ab=ba, cac-1=a-1b2, ad=da, eae-1=ab2, bc=cb, bd=db, be=eb, dcd-1=a2b2c, ce=ec, ede-1=b2d >

Subgroups: 252 in 190 conjugacy classes, 132 normal (52 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C42, C42, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, C2xC8, M4(2), C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4xC8, C4xC8, C8:C4, C22:C8, C4:C8, C4:C8, C2xC42, C42:C2, C4xD4, C4xD4, C4xQ8, C4:D4, C22:Q8, C22.D4, C4.4D4, C42.C2, C42:2C2, C22xC8, C22xC8, C2xM4(2), C2xC4xC8, C8o2M4(2), C42.6C4, C42.7C22, C8xD4, C8xD4, C8:9D4, C8:6D4, C8xQ8, C8:4Q8, C23.36C23, C42.291C23
Quotients: C1, C2, C4, C22, C2xC4, C23, C22xC4, C4oD4, C24, C8oD4, C23xC4, C2xC4oD4, C4xC4oD4, C2xC8oD4, C42.291C23

Smallest permutation representation of C42.291C23
On 64 points
Generators in S64
(1 33 20 48)(2 34 21 41)(3 35 22 42)(4 36 23 43)(5 37 24 44)(6 38 17 45)(7 39 18 46)(8 40 19 47)(9 51 28 60)(10 52 29 61)(11 53 30 62)(12 54 31 63)(13 55 32 64)(14 56 25 57)(15 49 26 58)(16 50 27 59)
(1 18 5 22)(2 19 6 23)(3 20 7 24)(4 21 8 17)(9 26 13 30)(10 27 14 31)(11 28 15 32)(12 29 16 25)(33 46 37 42)(34 47 38 43)(35 48 39 44)(36 41 40 45)(49 64 53 60)(50 57 54 61)(51 58 55 62)(52 59 56 63)
(1 48 5 44)(2 38 6 34)(3 42 7 46)(4 40 8 36)(9 64 13 60)(10 52 14 56)(11 58 15 62)(12 54 16 50)(17 41 21 45)(18 39 22 35)(19 43 23 47)(20 33 24 37)(25 57 29 61)(26 53 30 49)(27 59 31 63)(28 55 32 51)
(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 59 24 54)(2 64 17 51)(3 61 18 56)(4 58 19 53)(5 63 20 50)(6 60 21 55)(7 57 22 52)(8 62 23 49)(9 45 32 34)(10 42 25 39)(11 47 26 36)(12 44 27 33)(13 41 28 38)(14 46 29 35)(15 43 30 40)(16 48 31 37)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4N4O···4T8A···8H8I···8T8U···8AB
order1222222244444···44···48···88···88···8
size1111224411112···24···41···12···24···4

56 irreducible representations

dim11111111111111111222
type+++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C4C4C4C4C4C4C4oD4C8oD4C8oD4
kernelC42.291C23C2xC4xC8C8o2M4(2)C42.6C4C42.7C22C8xD4C8:9D4C8:6D4C8xQ8C8:4Q8C23.36C23C4:D4C22:Q8C22.D4C4.4D4C42.C2C42:2C2C8C4C22
# reps11212321111224224888

Matrix representation of C42.291C23 in GL4(F17) generated by

11600
21600
00015
0080
,
1000
0100
00130
00013
,
16000
15100
0002
0080
,
16100
15100
00013
00160
,
4000
0400
00015
0090
G:=sub<GL(4,GF(17))| [1,2,0,0,16,16,0,0,0,0,0,8,0,0,15,0],[1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,13],[16,15,0,0,0,1,0,0,0,0,0,8,0,0,2,0],[16,15,0,0,1,1,0,0,0,0,0,16,0,0,13,0],[4,0,0,0,0,4,0,0,0,0,0,9,0,0,15,0] >;

C42.291C23 in GAP, Magma, Sage, TeX

C_4^2._{291}C_2^3
% in TeX

G:=Group("C4^2.291C2^3");
// GroupNames label

G:=SmallGroup(128,1698);
// by ID

G=gap.SmallGroup(128,1698);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,2019,80,124]);
// Polycyclic

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

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