Copied to
clipboard

G = C168D4order 128 = 27

2nd semidirect product of C16 and D4 acting via D4/C22=C2

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

Aliases: C168D4, C221SD32, C23.28D8, C164C46C2, (C2×C4).61D8, C2.D162C2, (C2×C8).236D4, C8.101(C2×D4), C87D4.4C2, C2.9(C2×SD32), (C22×C16)⋊11C2, C2.Q322C2, (C2×SD32)⋊15C2, C4.19(C4○D8), C8.46(C4○D4), C8.18D42C2, (C2×D8).6C22, C2.12(C4○D16), C4.92(C4⋊D4), C2.20(C87D4), (C2×C8).522C23, C2.D8.7C22, (C2×C16).98C22, (C22×C4).592D4, C22.108(C2×D8), (C2×Q16).7C22, (C22×C8).532C22, (C2×C4).790(C2×D4), SmallGroup(128,949)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C168D4
C1C2C4C2×C4C2×C8C22×C8C22×C16 — C168D4
C1C2C4C2×C8 — C168D4
C1C22C22×C4C22×C8 — C168D4
C1C2C2C2C2C4C4C2×C8 — C168D4

Generators and relations for C168D4
 G = < a,b,c | a16=b4=c2=1, bab-1=cac=a7, cbc=b-1 >

Subgroups: 216 in 82 conjugacy classes, 34 normal (30 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×5], C8 [×2], C8, C2×C4 [×2], C2×C4 [×5], D4 [×4], Q8 [×2], C23, C23, C16 [×2], C16, C22⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×C8 [×2], D8 [×2], Q16 [×2], C22×C4, C2×D4 [×2], C2×Q8, D4⋊C4, Q8⋊C4, C2.D8 [×2], C2×C16 [×2], C2×C16 [×2], SD32 [×2], C4⋊D4, C22⋊Q8, C22×C8, C2×D8, C2×Q16, C2.D16, C2.Q32, C164C4, C87D4, C8.18D4, C22×C16, C2×SD32, C168D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], C2×D4 [×2], C4○D4, SD32 [×2], C4⋊D4, C2×D8, C4○D8, C87D4, C2×SD32, C4○D16, C168D4

Smallest permutation representation of C168D4
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 43 21 58)(2 34 22 49)(3 41 23 56)(4 48 24 63)(5 39 25 54)(6 46 26 61)(7 37 27 52)(8 44 28 59)(9 35 29 50)(10 42 30 57)(11 33 31 64)(12 40 32 55)(13 47 17 62)(14 38 18 53)(15 45 19 60)(16 36 20 51)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 25)(18 32)(19 23)(20 30)(22 28)(24 26)(27 31)(33 52)(34 59)(35 50)(36 57)(37 64)(38 55)(39 62)(40 53)(41 60)(42 51)(43 58)(44 49)(45 56)(46 63)(47 54)(48 61)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G8A···8H16A···16P
order122222244444448···816···16
size1111221622221616162···22···2

38 irreducible representations

dim11111111222222222
type+++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4C4○D4D8D8C4○D8SD32C4○D16
kernelC168D4C2.D16C2.Q32C164C4C87D4C8.18D4C22×C16C2×SD32C16C2×C8C22×C4C8C2×C4C23C4C22C2
# reps11111111211222488

Matrix representation of C168D4 in GL4(𝔽17) generated by

14300
141400
00615
0018
,
0400
4000
0048
00013
,
1000
01600
0010
001616
G:=sub<GL(4,GF(17))| [14,14,0,0,3,14,0,0,0,0,6,1,0,0,15,8],[0,4,0,0,4,0,0,0,0,0,4,0,0,0,8,13],[1,0,0,0,0,16,0,0,0,0,1,16,0,0,0,16] >;

C168D4 in GAP, Magma, Sage, TeX

C_{16}\rtimes_8D_4
% in TeX

G:=Group("C16:8D4");
// GroupNames label

G:=SmallGroup(128,949);
// by ID

G=gap.SmallGroup(128,949);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,736,422,1123,360,4037,124]);
// Polycyclic

G:=Group<a,b,c|a^16=b^4=c^2=1,b*a*b^-1=c*a*c=a^7,c*b*c=b^-1>;
// generators/relations

׿
×
𝔽