Copied to
clipboard

G = C89D8order 128 = 27

3rd semidirect product of C8 and D8 acting via D8/D4=C2

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

Aliases: C89D8, D41M4(2), C42.638C23, D4⋊C86C2, C2.4(C4×D8), (C8×D4)⋊34C2, (C4×D8).2C2, (C2×D8).6C4, C81C829C2, C8⋊C816C2, C4.86(C2×D8), C2.D8.9C4, C86D428C2, (C2×C8).377D4, D4⋊C4.3C4, C4.30(C8○D4), C2.8(C89D4), (C4×C8).38C22, (C4×D4).9C22, C4.128(C4○D8), C2.6(C8.26D4), C4⋊C8.220C22, C22.129(C4×D4), C4.24(C2×M4(2)), (C2×C8).30(C2×C4), C4⋊C4.132(C2×C4), (C2×D4).152(C2×C4), (C2×C4).1474(C2×D4), (C2×C4).499(C4○D4), (C2×C4).330(C22×C4), SmallGroup(128,313)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C89D8
C1C2C22C2×C4C42C4×C8C8×D4 — C89D8
C1C2C2×C4 — C89D8
C1C2×C4C4×C8 — C89D8
C1C22C22C42 — C89D8

Generators and relations for C89D8
 G = < a,b,c | a8=b8=c2=1, bab-1=cac=a5, cbc=b-1 >

Subgroups: 184 in 91 conjugacy classes, 44 normal (40 characteristic)
C1, C2 [×3], C2 [×3], C4 [×4], C4 [×3], C22, C22 [×7], C8 [×2], C8 [×6], C2×C4 [×3], C2×C4 [×6], D4 [×2], D4 [×3], C23 [×2], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×5], M4(2) [×2], D8 [×2], C22×C4 [×2], C2×D4 [×2], C4×C8 [×3], C22⋊C8 [×2], D4⋊C4 [×2], C4⋊C8 [×2], C2.D8, C4×D4 [×2], C22×C8, C2×M4(2), C2×D8, C8⋊C8, D4⋊C8 [×2], C81C8, C8×D4, C86D4, C4×D8, C89D8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, M4(2) [×2], D8 [×2], C22×C4, C2×D4, C4○D4, C4×D4, C2×M4(2), C8○D4, C2×D8, C4○D8, C89D4, C4×D8, C8.26D4, C89D8

Smallest permutation representation of C89D8
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 54 37 23 13 63 31 44)(2 51 38 20 14 60 32 41)(3 56 39 17 15 57 25 46)(4 53 40 22 16 62 26 43)(5 50 33 19 9 59 27 48)(6 55 34 24 10 64 28 45)(7 52 35 21 11 61 29 42)(8 49 36 18 12 58 30 47)
(1 13)(2 10)(3 15)(4 12)(5 9)(6 14)(7 11)(8 16)(17 56)(18 53)(19 50)(20 55)(21 52)(22 49)(23 54)(24 51)(26 30)(28 32)(34 38)(36 40)(41 64)(42 61)(43 58)(44 63)(45 60)(46 57)(47 62)(48 59)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E···8R8S8T
order12222224444444444488888···888
size11114481111222244822224···488

38 irreducible representations

dim11111111112222224
type+++++++++
imageC1C2C2C2C2C2C2C4C4C4D4D8C4○D4M4(2)C8○D4C4○D8C8.26D4
kernelC89D8C8⋊C8D4⋊C8C81C8C8×D4C86D4C4×D8D4⋊C4C2.D8C2×D8C2×C8C8C2×C4D4C4C4C2
# reps11211114222424442

Matrix representation of C89D8 in GL4(𝔽17) generated by

0100
4000
00130
00013
,
16000
0100
0006
001411
,
1000
01600
00160
0011
G:=sub<GL(4,GF(17))| [0,4,0,0,1,0,0,0,0,0,13,0,0,0,0,13],[16,0,0,0,0,1,0,0,0,0,0,14,0,0,6,11],[1,0,0,0,0,16,0,0,0,0,16,1,0,0,0,1] >;

C89D8 in GAP, Magma, Sage, TeX

C_8\rtimes_9D_8
% in TeX

G:=Group("C8:9D8");
// GroupNames label

G:=SmallGroup(128,313);
// by ID

G=gap.SmallGroup(128,313);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,100,1123,570,136,172]);
// Polycyclic

G:=Group<a,b,c|a^8=b^8=c^2=1,b*a*b^-1=c*a*c=a^5,c*b*c=b^-1>;
// generators/relations

׿
×
𝔽