Copied to
clipboard

G = C167D4order 128 = 27

1st semidirect product of C16 and D4 acting via D4/C22=C2

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

Aliases: C167D4, C221D16, C23.26D8, (C2×D16)⋊5C2, C87D42C2, C163C42C2, (C2×C4).59D8, C8.99(C2×D4), C2.6(C2×D16), C2.D161C2, (C22×C16)⋊8C2, (C2×C8).234D4, C8.44(C4○D4), C4.17(C4○D8), (C2×D8).5C22, C2.10(C4○D16), C2.18(C87D4), C4.90(C4⋊D4), (C2×C8).520C23, C2.D8.5C22, (C2×C16).81C22, C22.106(C2×D8), (C22×C4).590D4, (C22×C8).530C22, (C2×C4).788(C2×D4), SmallGroup(128,947)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C167D4
C1C2C4C2×C4C2×C8C22×C8C22×C16 — C167D4
C1C2C4C2×C8 — C167D4
C1C22C22×C4C22×C8 — C167D4
C1C2C2C2C2C4C4C2×C8 — C167D4

Generators and relations for C167D4
 G = < a,b,c | a16=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 264 in 87 conjugacy classes, 34 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×3], C22, C22 [×2], C22 [×8], C8 [×2], C8, C2×C4 [×2], C2×C4 [×4], D4 [×8], C23, C23 [×2], C16 [×2], C16, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], D8 [×4], C22×C4, C2×D4 [×4], D4⋊C4 [×2], C2.D8 [×2], C2×C16 [×2], C2×C16 [×2], D16 [×2], C4⋊D4 [×2], C22×C8, C2×D8 [×2], C2.D16 [×2], C163C4, C87D4 [×2], C22×C16, C2×D16, C167D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], C2×D4 [×2], C4○D4, D16 [×2], C4⋊D4, C2×D8, C4○D8, C87D4, C2×D16, C4○D16, C167D4

Smallest permutation representation of C167D4
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 48 20 59)(2 47 21 58)(3 46 22 57)(4 45 23 56)(5 44 24 55)(6 43 25 54)(7 42 26 53)(8 41 27 52)(9 40 28 51)(10 39 29 50)(11 38 30 49)(12 37 31 64)(13 36 32 63)(14 35 17 62)(15 34 18 61)(16 33 19 60)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 23)(18 22)(19 21)(24 32)(25 31)(26 30)(27 29)(33 58)(34 57)(35 56)(36 55)(37 54)(38 53)(39 52)(40 51)(41 50)(42 49)(43 64)(44 63)(45 62)(46 61)(47 60)(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), (1,48,20,59)(2,47,21,58)(3,46,22,57)(4,45,23,56)(5,44,24,55)(6,43,25,54)(7,42,26,53)(8,41,27,52)(9,40,28,51)(10,39,29,50)(11,38,30,49)(12,37,31,64)(13,36,32,63)(14,35,17,62)(15,34,18,61)(16,33,19,60), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,23)(18,22)(19,21)(24,32)(25,31)(26,30)(27,29)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,64)(44,63)(45,62)(46,61)(47,60)(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), (1,48,20,59)(2,47,21,58)(3,46,22,57)(4,45,23,56)(5,44,24,55)(6,43,25,54)(7,42,26,53)(8,41,27,52)(9,40,28,51)(10,39,29,50)(11,38,30,49)(12,37,31,64)(13,36,32,63)(14,35,17,62)(15,34,18,61)(16,33,19,60), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,23)(18,22)(19,21)(24,32)(25,31)(26,30)(27,29)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,64)(44,63)(45,62)(46,61)(47,60)(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)], [(1,48,20,59),(2,47,21,58),(3,46,22,57),(4,45,23,56),(5,44,24,55),(6,43,25,54),(7,42,26,53),(8,41,27,52),(9,40,28,51),(10,39,29,50),(11,38,30,49),(12,37,31,64),(13,36,32,63),(14,35,17,62),(15,34,18,61),(16,33,19,60)], [(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(17,23),(18,22),(19,21),(24,32),(25,31),(26,30),(27,29),(33,58),(34,57),(35,56),(36,55),(37,54),(38,53),(39,52),(40,51),(41,50),(42,49),(43,64),(44,63),(45,62),(46,61),(47,60),(48,59)])

38 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F8A···8H16A···16P
order122222224444448···816···16
size1111221616222216162···22···2

38 irreducible representations

dim111111222222222
type++++++++++++
imageC1C2C2C2C2C2D4D4D4C4○D4D8D8C4○D8D16C4○D16
kernelC167D4C2.D16C163C4C87D4C22×C16C2×D16C16C2×C8C22×C4C8C2×C4C23C4C22C2
# reps121211211222488

Matrix representation of C167D4 in GL4(𝔽17) generated by

10800
13200
00613
0046
,
13900
0400
00160
0001
,
1000
161600
0010
00016
G:=sub<GL(4,GF(17))| [10,13,0,0,8,2,0,0,0,0,6,4,0,0,13,6],[13,0,0,0,9,4,0,0,0,0,16,0,0,0,0,1],[1,16,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;

C167D4 in GAP, Magma, Sage, TeX

C_{16}\rtimes_7D_4
% in TeX

G:=Group("C16:7D4");
// GroupNames label

G:=SmallGroup(128,947);
// by ID

G=gap.SmallGroup(128,947);
# by ID

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

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

׿
×
𝔽