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G = C42.264D4order 128 = 27

246th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.264D4, C42.396C23, C4.982+ 1+4, (C2×C4)⋊6SD16, C85D420C2, Q8⋊D430C2, C4.71(C2×SD16), D4.D437C2, C4⋊C8.336C22, C4⋊C4.145C23, (C4×C8).264C22, (C2×C4).404C24, (C2×C8).324C23, C4.SD1625C2, (C22×C4).494D4, C23.689(C2×D4), C4⋊Q8.299C22, (C4×D4).103C22, (C2×D4).154C23, C4.25(C8.C22), (C2×Q8).141C23, C2.22(C22×SD16), C22.38(C2×SD16), C42.12C442C2, C4⋊D4.187C22, C41D4.160C22, C22⋊C8.218C22, (C2×C42).871C22, Q8⋊C4.99C22, (C2×SD16).83C22, C22.664(C22×D4), (C22×C4).1075C23, (C22×Q8).319C22, C2.75(C22.29C24), C22.26C24.40C2, (C2×C4⋊Q8)⋊40C2, (C2×C4).865(C2×D4), C2.53(C2×C8.C22), SmallGroup(128,1938)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.264D4
C1C2C4C2×C4C22×C4C22×Q8C2×C4⋊Q8 — C42.264D4
C1C2C2×C4 — C42.264D4
C1C22C2×C42 — C42.264D4
C1C2C2C2×C4 — C42.264D4

Generators and relations for C42.264D4
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, ac=ca, dad=a-1, cbc-1=a2b-1, dbd=a2b, dcd=c3 >

Subgroups: 444 in 214 conjugacy classes, 96 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×8], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×18], D4 [×12], Q8 [×14], C23, C23 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×9], C2×C8 [×4], SD16 [×8], C22×C4 [×3], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×4], C2×Q8 [×7], C4○D4 [×4], C4×C8 [×2], C22⋊C8 [×2], Q8⋊C4 [×8], C4⋊C8 [×2], C2×C42, C2×C4⋊C4 [×2], C4×D4 [×2], C4×D4, C4⋊D4 [×2], C4⋊D4, C4.4D4, C41D4, C4⋊Q8, C4⋊Q8 [×4], C4⋊Q8 [×2], C2×SD16 [×8], C22×Q8 [×2], C2×C4○D4, C42.12C4, Q8⋊D4 [×4], D4.D4 [×4], C4.SD16 [×2], C85D4 [×2], C2×C4⋊Q8, C22.26C24, C42.264D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], SD16 [×4], C2×D4 [×6], C24, C2×SD16 [×6], C8.C22 [×2], C22×D4, 2+ 1+4 [×2], C22.29C24, C22×SD16, C2×C8.C22, C42.264D4

Smallest permutation representation of C42.264D4
On 64 points
Generators in S64
(1 63 25 55)(2 64 26 56)(3 57 27 49)(4 58 28 50)(5 59 29 51)(6 60 30 52)(7 61 31 53)(8 62 32 54)(9 18 40 47)(10 19 33 48)(11 20 34 41)(12 21 35 42)(13 22 36 43)(14 23 37 44)(15 24 38 45)(16 17 39 46)
(1 37 5 33)(2 11 6 15)(3 39 7 35)(4 13 8 9)(10 25 14 29)(12 27 16 31)(17 53 21 49)(18 58 22 62)(19 55 23 51)(20 60 24 64)(26 34 30 38)(28 36 32 40)(41 52 45 56)(42 57 46 61)(43 54 47 50)(44 59 48 63)
(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)
(2 4)(3 7)(6 8)(9 38)(10 33)(11 36)(12 39)(13 34)(14 37)(15 40)(16 35)(17 21)(18 24)(20 22)(26 28)(27 31)(30 32)(41 43)(42 46)(45 47)(49 61)(50 64)(51 59)(52 62)(53 57)(54 60)(55 63)(56 58)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4H4I4J4K···4P8A···8H
order122222224···4444···48···8
size111122882···2448···84···4

32 irreducible representations

dim1111111122244
type++++++++++-+
imageC1C2C2C2C2C2C2C2D4D4SD16C8.C222+ 1+4
kernelC42.264D4C42.12C4Q8⋊D4D4.D4C4.SD16C85D4C2×C4⋊Q8C22.26C24C42C22×C4C2×C4C4C4
# reps1144221122822

Matrix representation of C42.264D4 in GL6(𝔽17)

010000
1600000
000106
001060
00011016
00110160
,
0160000
100000
000100
001000
000001
000010
,
1250000
12120000
000010
0000016
001000
0001600
,
100000
0160000
001000
0001600
000010
0000016

G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,11,0,0,1,0,11,0,0,0,0,6,0,16,0,0,6,0,16,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,12,0,0,0,0,5,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,16,0,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;

C42.264D4 in GAP, Magma, Sage, TeX

C_4^2._{264}D_4
% in TeX

G:=Group("C4^2.264D4");
// GroupNames label

G:=SmallGroup(128,1938);
// by ID

G=gap.SmallGroup(128,1938);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,219,352,675,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,a*c=c*a,d*a*d=a^-1,c*b*c^-1=a^2*b^-1,d*b*d=a^2*b,d*c*d=c^3>;
// generators/relations

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