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G = C8.21D8order 128 = 27

12nd non-split extension by C8 of D8 acting via D8/C8=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C8.21D8, C16.12D4, C42.339D4, (C4×C16)⋊11C2, (C2×Q32)⋊6C2, C8.42(C2×D4), (C2×C4).63D8, C4.10(C2×D8), (C2×D16).3C2, (C2×C8).280D4, (C2×SD32)⋊17C2, C8.12D42C2, C4.4(C41D4), C2.18(C4○D16), C2.15(C84D4), (C4×C8).414C22, (C2×C8).547C23, (C2×C16).85C22, (C2×D8).18C22, C22.133(C2×D8), (C2×Q16).19C22, (C2×C4).815(C2×D4), SmallGroup(128,981)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C8.21D8
C1C2C4C2×C4C2×C8C4×C8C4×C16 — C8.21D8
C1C2C4C2×C8 — C8.21D8
C1C22C42C4×C8 — C8.21D8
C1C2C2C2C2C4C4C2×C8 — C8.21D8

Generators and relations for C8.21D8
 G = < a,b,c | a8=c2=1, b8=a4, ab=ba, cac=a3, cbc=a4b7 >

Subgroups: 264 in 92 conjugacy classes, 36 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×4], Q8 [×4], C23 [×2], C16 [×4], C42, C22⋊C4 [×4], C2×C8 [×2], D8 [×4], SD16 [×4], Q16 [×4], C2×D4 [×2], C2×Q8 [×2], C4×C8, C2×C16 [×2], D16 [×2], SD32 [×4], Q32 [×2], C4.4D4 [×2], C2×D8 [×2], C2×SD16 [×2], C2×Q16 [×2], C4×C16, C8.12D4 [×2], C2×D16, C2×SD32 [×2], C2×Q32, C8.21D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×4], C2×D4 [×3], C41D4, C2×D8 [×2], C84D4, C4○D16 [×2], C8.21D8

Smallest permutation representation of C8.21D8
On 64 points
Generators in S64
(1 18 51 46 9 26 59 38)(2 19 52 47 10 27 60 39)(3 20 53 48 11 28 61 40)(4 21 54 33 12 29 62 41)(5 22 55 34 13 30 63 42)(6 23 56 35 14 31 64 43)(7 24 57 36 15 32 49 44)(8 25 58 37 16 17 50 45)
(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 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 46)(18 45)(19 44)(20 43)(21 42)(22 41)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)(31 48)(32 47)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 64)(62 63)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E4A···4F4G4H8A···8H16A···16P
order1222224···4448···816···16
size111116162···216162···22···2

38 irreducible representations

dim111111222222
type+++++++++++
imageC1C2C2C2C2C2D4D4D4D8D8C4○D16
kernelC8.21D8C4×C16C8.12D4C2×D16C2×SD32C2×Q32C16C42C2×C8C8C2×C4C2
# reps1121214114416

Matrix representation of C8.21D8 in GL4(𝔽17) generated by

161500
1100
00107
0050
,
16000
01600
00712
00112
,
16000
1100
00712
001310
G:=sub<GL(4,GF(17))| [16,1,0,0,15,1,0,0,0,0,10,5,0,0,7,0],[16,0,0,0,0,16,0,0,0,0,7,11,0,0,12,2],[16,1,0,0,0,1,0,0,0,0,7,13,0,0,12,10] >;

C8.21D8 in GAP, Magma, Sage, TeX

C_8._{21}D_8
% in TeX

G:=Group("C8.21D8");
// GroupNames label

G:=SmallGroup(128,981);
// by ID

G=gap.SmallGroup(128,981);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,288,422,436,1123,360,4037,124]);
// Polycyclic

G:=Group<a,b,c|a^8=c^2=1,b^8=a^4,a*b=b*a,c*a*c=a^3,c*b*c=a^4*b^7>;
// generators/relations

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