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G = C42.290C23order 128 = 27

151st non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.290C23, C8:13(C4oD4), C4o2(C8:6D4), C4o2(C8:9D4), C8:6D4:51C2, C8:9D4:53C2, C4o2(C8:4Q8), C8:4Q8:51C2, (C4xD4).29C4, (C2xC4):9M4(2), (C4xQ8).27C4, C4.64(C8oD4), C42o(C8:9D4), C42o(C8:6D4), C42o(C8:4Q8), C4:C8.361C22, (C4xM4(2)):35C2, (C4xC8).437C22, (C2xC8).427C23, (C2xC4).662C24, C42.249(C2xC4), C4.14(C2xM4(2)), C42:C2.31C4, (C4xD4).293C22, (C4xQ8).278C22, C8:C4.162C22, C42.12C4:49C2, C22.3(C2xM4(2)), C22:C8.232C22, C22.187(C23xC4), (C22xC8).585C22, (C22xC4).934C23, C23.146(C22xC4), C2.15(C22xM4(2)), (C2xC42).1119C22, (C2xM4(2)).364C22, (C2xC4xC8):44C2, C2.44(C4xC4oD4), C2.23(C2xC8oD4), C4:C4.225(C2xC4), (C2xC4oD4).25C4, (C4xC4oD4).15C2, C4.313(C2xC4oD4), (C2xD4).232(C2xC4), C22:C4.74(C2xC4), (C2xQ8).209(C2xC4), (C22xC4).420(C2xC4), (C2xC4).271(C22xC4), SmallGroup(128,1697)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.290C23
C1C2C4C2xC4C42C4xC8C2xC4xC8 — C42.290C23
C1C22 — C42.290C23
C1C42 — C42.290C23
C1C2C2C2xC4 — C42.290C23

Generators and relations for C42.290C23
 G = < a,b,c,d,e | a4=b4=c2=e2=1, d2=b-1, ab=ba, cac=a-1, dad-1=ab2, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=a2c, ede=b2d >

Subgroups: 276 in 206 conjugacy classes, 140 normal (36 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C42, C42, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, M4(2), C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C4xC8, C4xC8, C8:C4, C22:C8, C4:C8, C2xC42, C2xC42, C42:C2, C42:C2, C4xD4, C4xD4, C4xQ8, C22xC8, C2xM4(2), C2xC4oD4, C2xC4xC8, C4xM4(2), C42.12C4, C42.12C4, C8:9D4, C8:6D4, C8:4Q8, C4xC4oD4, C42.290C23
Quotients: C1, C2, C4, C22, C2xC4, C23, M4(2), C22xC4, C4oD4, C24, C2xM4(2), C8oD4, C23xC4, C2xC4oD4, C4xC4oD4, C22xM4(2), C2xC8oD4, C42.290C23

Smallest permutation representation of C42.290C23
On 64 points
Generators in S64
(1 38 25 12)(2 35 26 9)(3 40 27 14)(4 37 28 11)(5 34 29 16)(6 39 30 13)(7 36 31 10)(8 33 32 15)(17 47 55 60)(18 44 56 57)(19 41 49 62)(20 46 50 59)(21 43 51 64)(22 48 52 61)(23 45 53 58)(24 42 54 63)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)(33 39 37 35)(34 40 38 36)(41 47 45 43)(42 48 46 44)(49 55 53 51)(50 56 54 52)(57 63 61 59)(58 64 62 60)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 41)(7 42)(8 43)(9 53)(10 54)(11 55)(12 56)(13 49)(14 50)(15 51)(16 52)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(25 57)(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(32 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 5)(3 7)(10 14)(12 16)(17 55)(18 52)(19 49)(20 54)(21 51)(22 56)(23 53)(24 50)(25 29)(27 31)(34 38)(36 40)(41 62)(42 59)(43 64)(44 61)(45 58)(46 63)(47 60)(48 57)

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

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

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

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

dim111111111111222
type++++++++
imageC1C2C2C2C2C2C2C2C4C4C4C4C4oD4M4(2)C8oD4
kernelC42.290C23C2xC4xC8C4xM4(2)C42.12C4C8:9D4C8:6D4C8:4Q8C4xC4oD4C42:C2C4xD4C4xQ8C2xC4oD4C8C2xC4C4
# reps112342216622888

Matrix representation of C42.290C23 in GL4(F17) generated by

4000
01300
0010
00116
,
1000
0100
00130
00013
,
0100
1000
0010
0001
,
1000
0100
0049
001013
,
1000
01600
00160
00161
G:=sub<GL(4,GF(17))| [4,0,0,0,0,13,0,0,0,0,1,1,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,13],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,4,10,0,0,9,13],[1,0,0,0,0,16,0,0,0,0,16,16,0,0,0,1] >;

C42.290C23 in GAP, Magma, Sage, TeX

C_4^2._{290}C_2^3
% in TeX

G:=Group("C4^2.290C2^3");
// GroupNames label

G:=SmallGroup(128,1697);
// by ID

G=gap.SmallGroup(128,1697);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,184,80,124]);
// Polycyclic

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

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