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G = C42.261C23order 128 = 27

122nd non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.261C23, (C4×D4).24C4, (C4×Q8).23C4, C4⋊C8.230C22, (C4×M4(2))⋊32C2, (C2×C8).399C23, C42.205(C2×C4), (C4×C8).326C22, (C2×C4).642C24, C42.6C444C2, C8⋊C4.153C22, C4.17(C42⋊C2), C2.11(Q8○M4(2)), C22⋊C8.138C22, C22.170(C23×C4), (C2×C42).755C22, (C22×C8).430C22, C23.101(C22×C4), (C22×C4).913C23, C42.7C2220C2, C42.6C2228C2, C22.5(C42⋊C2), C42⋊C2.291C22, (C2×M4(2)).344C22, (C2×C8⋊C4)⋊32C2, C4⋊C4.218(C2×C4), (C4×C4○D4).12C2, C4.293(C2×C4○D4), (C2×D4).228(C2×C4), C22⋊C4.69(C2×C4), (C2×Q8).206(C2×C4), (C2×C4).680(C4○D4), (C2×C4).258(C22×C4), (C22×C4).336(C2×C4), C2.42(C2×C42⋊C2), (C22×C8)⋊C2.18C2, (C2×C4○D4).281C22, SmallGroup(128,1655)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.261C23
C1C2C4C2×C4C22×C4C2×C42C4×C4○D4 — C42.261C23
C1C22 — C42.261C23
C1C2×C4 — C42.261C23
C1C2C2C2×C4 — C42.261C23

Generators and relations for C42.261C23
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=a-1b2, ad=da, eae=ab2, bc=cb, bd=db, be=eb, dcd=a2c, ce=ec, ede=b2d >

Subgroups: 268 in 192 conjugacy classes, 132 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×2], C22 [×8], C8 [×8], C2×C4 [×2], C2×C4 [×14], C2×C4 [×10], D4 [×6], Q8 [×2], C23, C23 [×2], C42 [×2], C42 [×8], C22⋊C4 [×6], C4⋊C4 [×6], C2×C8 [×8], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4 [×8], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], C4×C8 [×2], C8⋊C4 [×6], C22⋊C8 [×8], C4⋊C8 [×8], C2×C42, C2×C42 [×2], C42⋊C2, C42⋊C2 [×2], C4×D4 [×6], C4×Q8 [×2], C22×C8 [×2], C2×M4(2) [×2], C2×C4○D4, C2×C8⋊C4, C4×M4(2), (C22×C8)⋊C2 [×2], C42.6C22 [×2], C42.6C4 [×4], C42.7C22 [×4], C4×C4○D4, C42.261C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×4], C24, C42⋊C2 [×4], C23×C4, C2×C4○D4 [×2], C2×C42⋊C2, Q8○M4(2) [×2], C42.261C23

Smallest permutation representation of C42.261C23
On 64 points
Generators in S64
(1 33 55 18)(2 23 56 38)(3 35 49 20)(4 17 50 40)(5 37 51 22)(6 19 52 34)(7 39 53 24)(8 21 54 36)(9 44 28 58)(10 63 29 41)(11 46 30 60)(12 57 31 43)(13 48 32 62)(14 59 25 45)(15 42 26 64)(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)(2,23,56,38)(3,35,49,20)(4,17,50,40)(5,37,51,22)(6,19,52,34)(7,39,53,24)(8,21,54,36)(9,44,28,58)(10,63,29,41)(11,46,30,60)(12,57,31,43)(13,48,32,62)(14,59,25,45)(15,42,26,64)(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)(2,23,56,38)(3,35,49,20)(4,17,50,40)(5,37,51,22)(6,19,52,34)(7,39,53,24)(8,21,54,36)(9,44,28,58)(10,63,29,41)(11,46,30,60)(12,57,31,43)(13,48,32,62)(14,59,25,45)(15,42,26,64)(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),(2,23,56,38),(3,35,49,20),(4,17,50,40),(5,37,51,22),(6,19,52,34),(7,39,53,24),(8,21,54,36),(9,44,28,58),(10,63,29,41),(11,46,30,60),(12,57,31,43),(13,48,32,62),(14,59,25,45),(15,42,26,64),(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)])

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4N4O···4T8A···8P
order1222222244444···44···48···8
size1111224411112···24···44···4

44 irreducible representations

dim111111111124
type++++++++
imageC1C2C2C2C2C2C2C2C4C4C4○D4Q8○M4(2)
kernelC42.261C23C2×C8⋊C4C4×M4(2)(C22×C8)⋊C2C42.6C22C42.6C4C42.7C22C4×C4○D4C4×D4C4×Q8C2×C4C2
# reps1112244112484

Matrix representation of C42.261C23 in GL6(𝔽17)

1300000
640000
0000013
000040
000400
0013000
,
1600000
0160000
0013000
0001300
0000130
0000013
,
14130000
1130000
0001007
0010070
000707
007070
,
1600000
1010000
0001300
004000
0000013
000040
,
100000
010000
000100
001000
000001
000010

G:=sub<GL(6,GF(17))| [13,6,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,13,0,0,0,0,4,0,0,0,0,4,0,0,0,0,13,0,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[14,11,0,0,0,0,13,3,0,0,0,0,0,0,0,10,0,7,0,0,10,0,7,0,0,0,0,7,0,7,0,0,7,0,7,0],[16,10,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,13,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C42.261C23 in GAP, Magma, Sage, TeX

C_4^2._{261}C_2^3
% in TeX

G:=Group("C4^2.261C2^3");
// GroupNames label

G:=SmallGroup(128,1655);
// by ID

G=gap.SmallGroup(128,1655);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,100,1018,124]);
// 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*b^2,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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