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G = C42.301C23order 128 = 27

162nd non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.301C23, C4.1752+ 1+4, (C8×D4)⋊49C2, C4⋊Q8.34C4, C86D444C2, C89D445C2, C4.21(C8○D4), C4⋊D4.29C4, C41D4.21C4, (C4×C8).29C22, C4⋊C8.367C22, C42.226(C2×C4), (C2×C4).678C24, (C2×C8).439C23, C4.4D4.22C4, C8⋊C4.98C22, C42.6C453C2, (C4×D4).303C22, C23.45(C22×C4), C42.12C455C2, C2.32(Q8○M4(2)), C22⋊C8.146C22, (C22×C8).451C22, (C2×C42).785C22, (C22×C4).945C23, C22.202(C23×C4), C2.52(C22.11C24), (C2×M4(2)).248C22, C22.26C24.29C2, C2.31(C2×C8○D4), C4⋊C4.121(C2×C4), (C2×D4).145(C2×C4), C22⋊C4.22(C2×C4), (C2×C4).83(C22×C4), (C2×Q8).127(C2×C4), (C22×C8)⋊C237C2, (C22×C4).358(C2×C4), (C2×C4○D4).98C22, SmallGroup(128,1713)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.301C23
C1C2C4C2×C4C22×C4C2×C4○D4C22.26C24 — C42.301C23
C1C22 — C42.301C23
C1C2×C4 — C42.301C23
C1C2C2C2×C4 — C42.301C23

Generators and relations for C42.301C23
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=a-1b2, dad=a-1, ae=ea, bc=cb, bd=db, be=eb, dcd=ece=a2b2c, ede=b2d >

Subgroups: 332 in 201 conjugacy classes, 126 normal (28 characteristic)
C1, C2 [×3], C2 [×5], C4 [×4], C4 [×8], C22, C22 [×15], C8 [×8], C2×C4 [×6], C2×C4 [×4], C2×C4 [×13], D4 [×12], Q8 [×2], C23, C23 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×4], M4(2) [×4], C22×C4 [×3], C22×C4 [×4], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×4], C4×C8 [×2], C8⋊C4 [×2], C22⋊C8 [×12], C4⋊C8 [×4], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×2], C41D4, C4⋊Q8, C22×C8 [×4], C2×M4(2) [×4], C2×C4○D4 [×2], (C22×C8)⋊C2 [×4], C42.12C4, C42.6C4, C8×D4 [×2], C89D4 [×4], C86D4 [×2], C22.26C24, C42.301C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C8○D4 [×2], C23×C4, 2+ 1+4 [×2], C22.11C24, C2×C8○D4, Q8○M4(2), C42.301C23

Smallest permutation representation of C42.301C23
On 64 points
Generators in S64
(1 12 51 37)(2 34 52 9)(3 14 53 39)(4 36 54 11)(5 16 55 33)(6 38 56 13)(7 10 49 35)(8 40 50 15)(17 43 29 63)(18 60 30 48)(19 45 31 57)(20 62 32 42)(21 47 25 59)(22 64 26 44)(23 41 27 61)(24 58 28 46)
(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 21)(2 30)(3 23)(4 32)(5 17)(6 26)(7 19)(8 28)(9 48)(10 57)(11 42)(12 59)(13 44)(14 61)(15 46)(16 63)(18 52)(20 54)(22 56)(24 50)(25 51)(27 53)(29 55)(31 49)(33 43)(34 60)(35 45)(36 62)(37 47)(38 64)(39 41)(40 58)
(1 23)(2 32)(3 17)(4 26)(5 19)(6 28)(7 21)(8 30)(9 62)(10 47)(11 64)(12 41)(13 58)(14 43)(15 60)(16 45)(18 50)(20 52)(22 54)(24 56)(25 49)(27 51)(29 53)(31 55)(33 57)(34 42)(35 59)(36 44)(37 61)(38 46)(39 63)(40 48)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D···2H4A4B4C4D4E4F4G4H4I···4O8A···8H8I···8T
order12222···2444444444···48···88···8
size11114···4111122224···42···24···4

44 irreducible representations

dim111111111111244
type+++++++++
imageC1C2C2C2C2C2C2C2C4C4C4C4C8○D42+ 1+4Q8○M4(2)
kernelC42.301C23(C22×C8)⋊C2C42.12C4C42.6C4C8×D4C89D4C86D4C22.26C24C4⋊D4C4.4D4C41D4C4⋊Q8C4C4C2
# reps141124218422822

Matrix representation of C42.301C23 in GL6(𝔽17)

0130000
1300000
0000016
0000160
000100
001000
,
400000
040000
001000
000100
000010
000001
,
200000
020000
0016000
000100
0000160
000001
,
040000
1300000
000100
001000
0000016
0000160
,
010000
100000
000100
001000
000001
000010

G:=sub<GL(6,GF(17))| [0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16,0,0,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C42.301C23 in GAP, Magma, Sage, TeX

C_4^2._{301}C_2^3
% in TeX

G:=Group("C4^2.301C2^3");
// GroupNames label

G:=SmallGroup(128,1713);
// by ID

G=gap.SmallGroup(128,1713);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,891,675,1018,521,124]);
// 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,d*a*d=a^-1,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=e*c*e=a^2*b^2*c,e*d*e=b^2*d>;
// generators/relations

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