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G = C42.530C23order 128 = 27

391st non-split extension by C42 of C23 acting via C23/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C42.530C23, C4.1472+ 1+4, (C4×D8)⋊26C2, (C8×Q8)⋊18C2, C4⋊C4.288D4, (C4×Q16)⋊18C2, C4⋊SD1646C2, C2.71(D4○D8), C8.96(C4○D4), C4.94(C4○D8), C84D4.11C2, (C2×Q8).191D4, D4.2D411C2, C8.12D411C2, C4⋊C4.447C23, C4⋊C8.332C22, (C2×C4).588C24, (C4×C8).128C22, (C2×C8).221C23, (C2×D8).42C22, C2.42(Q86D4), (C4×D4).221C22, (C2×D4).282C23, (C4×Q8).317C22, (C2×Q8).267C23, C2.D8.239C22, C41D4.107C22, (C2×Q16).145C22, C4.4D4.88C22, C22.848(C22×D4), D4⋊C4.192C22, C22.53C247C2, Q8⋊C4.168C22, (C2×SD16).104C22, C2.81(C2×C4○D8), C4.166(C2×C4○D4), (C2×C4).183(C2×D4), SmallGroup(128,2128)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.530C23
C1C2C4C2×C4C42C4×Q8C22.53C24 — C42.530C23
C1C2C2×C4 — C42.530C23
C1C22C4×Q8 — C42.530C23
C1C2C2C2×C4 — C42.530C23

Generators and relations for C42.530C23
 G = < a,b,c,d,e | a4=b4=1, c2=b2, d2=a2, e2=a2b2, ab=ba, cac-1=eae-1=a-1b2, ad=da, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ce=ec, de=ed >

Subgroups: 408 in 192 conjugacy classes, 88 normal (24 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×12], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], D4 [×14], Q8 [×6], C23 [×4], C42, C42 [×2], C42 [×2], C22⋊C4 [×12], C4⋊C4, C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], D8 [×6], SD16 [×4], Q16 [×2], C22×C4 [×4], C2×D4 [×4], C2×D4 [×4], C2×Q8, C2×Q8 [×2], C4×C8, C4×C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×2], C4⋊C8, C4⋊C8 [×2], C2.D8, C4×D4 [×4], C4×D4 [×2], C4×Q8, C4×Q8 [×2], C22.D4 [×4], C4.4D4 [×4], C4.4D4 [×2], C41D4 [×2], C2×D8 [×4], C2×SD16 [×4], C2×Q16, C4×D8 [×2], C4×Q16, C8×Q8, C4⋊SD16 [×2], D4.2D4 [×4], C84D4, C8.12D4 [×2], C22.53C24 [×2], C42.530C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C4○D8 [×2], C22×D4, C2×C4○D4, 2+ 1+4, Q86D4, C2×C4○D8, D4○D8, C42.530C23

Smallest permutation representation of C42.530C23
On 64 points
Generators in S64
(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 53 12 26)(2 54 9 27)(3 55 10 28)(4 56 11 25)(5 43 39 23)(6 44 40 24)(7 41 37 21)(8 42 38 22)(13 31 50 60)(14 32 51 57)(15 29 52 58)(16 30 49 59)(17 62 48 33)(18 63 45 34)(19 64 46 35)(20 61 47 36)
(1 49 12 16)(2 15 9 52)(3 51 10 14)(4 13 11 50)(5 17 39 48)(6 47 40 20)(7 19 37 46)(8 45 38 18)(21 64 41 35)(22 34 42 63)(23 62 43 33)(24 36 44 61)(25 31 56 60)(26 59 53 30)(27 29 54 58)(28 57 55 32)
(1 45 3 47)(2 46 4 48)(5 58 7 60)(6 59 8 57)(9 19 11 17)(10 20 12 18)(13 23 15 21)(14 24 16 22)(25 33 27 35)(26 34 28 36)(29 37 31 39)(30 38 32 40)(41 50 43 52)(42 51 44 49)(53 63 55 61)(54 64 56 62)
(1 15 10 50)(2 51 11 16)(3 13 12 52)(4 49 9 14)(5 63 37 36)(6 33 38 64)(7 61 39 34)(8 35 40 62)(17 22 46 44)(18 41 47 23)(19 24 48 42)(20 43 45 21)(25 30 54 57)(26 58 55 31)(27 32 56 59)(28 60 53 29)

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

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

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

35 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4H4I···4O4P4Q8A8B8C8D8E···8J
order122222224···44···44488888···8
size111188882···24···48822224···4

35 irreducible representations

dim111111111222244
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4C4○D4C4○D82+ 1+4D4○D8
kernelC42.530C23C4×D8C4×Q16C8×Q8C4⋊SD16D4.2D4C84D4C8.12D4C22.53C24C4⋊C4C2×Q8C8C4C4C2
# reps121124122314812

Matrix representation of C42.530C23 in GL4(𝔽17) generated by

13000
01300
0040
001513
,
161500
1100
0010
0001
,
4000
131300
00131
0024
,
0700
12000
00130
00013
,
1000
0100
0014
00816
G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,4,15,0,0,0,13],[16,1,0,0,15,1,0,0,0,0,1,0,0,0,0,1],[4,13,0,0,0,13,0,0,0,0,13,2,0,0,1,4],[0,12,0,0,7,0,0,0,0,0,13,0,0,0,0,13],[1,0,0,0,0,1,0,0,0,0,1,8,0,0,4,16] >;

C42.530C23 in GAP, Magma, Sage, TeX

C_4^2._{530}C_2^3
% in TeX

G:=Group("C4^2.530C2^3");
// GroupNames label

G:=SmallGroup(128,2128);
// by ID

G=gap.SmallGroup(128,2128);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,568,758,436,346,80,4037,1027,124]);
// Polycyclic

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

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