Copied to
clipboard

G = C4×SD32order 128 = 27

Direct product of C4 and SD32

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

Aliases: C4×SD32, C42.330D4, C168(C2×C4), (C4×C16)⋊12C2, Q162(C2×C4), (C4×Q16)⋊1C2, (C4×D8).4C2, D8.2(C2×C4), C2.14(C4×D8), C4.26(C4×D4), C42(C164C4), C164C414C2, (C2×C8).232D4, (C2×C4).173D8, C43(C2.D16), C2.4(C2×SD32), C2.D16.8C2, C2.4(C4○D16), C8.39(C4○D4), C4.12(C4○D8), C8.36(C22×C4), (C2×SD32).5C2, C22.62(C2×D8), C42(C2.Q32), C2.Q3221C2, (C2×C16).70C22, (C2×C8).503C23, (C4×C8).397C22, (C2×D8).104C22, C2.D8.151C22, (C2×Q16).102C22, (C2×C4).769(C2×D4), SmallGroup(128,905)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C4×SD32
C1C2C4C2×C4C2×C8C4×C8C4×Q16 — C4×SD32
C1C2C4C8 — C4×SD32
C1C2×C4C42C4×C8 — C4×SD32
C1C2C2C2C2C4C4C2×C8 — C4×SD32

Generators and relations for C4×SD32
 G = < a,b,c | a4=b16=c2=1, ab=ba, ac=ca, cbc=b7 >

Subgroups: 196 in 81 conjugacy classes, 42 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×5], C22, C22 [×4], C8 [×2], C8, C2×C4 [×3], C2×C4 [×5], D4 [×3], Q8 [×3], C23, C16 [×2], C16, C42, C42, C22⋊C4, C4⋊C4 [×3], C2×C8 [×2], D8 [×2], D8, Q16 [×2], Q16, C22×C4, C2×D4, C2×Q8, C4×C8, D4⋊C4, Q8⋊C4, C2.D8 [×2], C2×C16 [×2], SD32 [×4], C4×D4, C4×Q8, C2×D8, C2×Q16, C4×C16, C2.D16, C2.Q32, C164C4, C4×D8, C4×Q16, C2×SD32, C4×SD32
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D8 [×2], C22×C4, C2×D4, C4○D4, SD32 [×2], C4×D4, C2×D8, C4○D8, C4×D8, C2×SD32, C4○D16, C4×SD32

Smallest permutation representation of C4×SD32
On 64 points
Generators in S64
(1 38 27 62)(2 39 28 63)(3 40 29 64)(4 41 30 49)(5 42 31 50)(6 43 32 51)(7 44 17 52)(8 45 18 53)(9 46 19 54)(10 47 20 55)(11 48 21 56)(12 33 22 57)(13 34 23 58)(14 35 24 59)(15 36 25 60)(16 37 26 61)
(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 27)(2 18)(3 25)(4 32)(5 23)(6 30)(7 21)(8 28)(9 19)(10 26)(11 17)(12 24)(13 31)(14 22)(15 29)(16 20)(33 59)(34 50)(35 57)(36 64)(37 55)(38 62)(39 53)(40 60)(41 51)(42 58)(43 49)(44 56)(45 63)(46 54)(47 61)(48 52)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I···4N8A···8H16A···16P
order122222444444444···48···816···16
size111188111122228···82···22···2

44 irreducible representations

dim1111111112222222
type+++++++++++
imageC1C2C2C2C2C2C2C2C4D4D4C4○D4D8SD32C4○D8C4○D16
kernelC4×SD32C4×C16C2.D16C2.Q32C164C4C4×D8C4×Q16C2×SD32SD32C42C2×C8C8C2×C4C4C4C2
# reps1111111181124848

Matrix representation of C4×SD32 in GL4(𝔽17) generated by

13000
01300
0040
0004
,
141400
31400
0017
00101
,
16000
0100
00160
0001
G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,4,0,0,0,0,4],[14,3,0,0,14,14,0,0,0,0,1,10,0,0,7,1],[16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,1] >;

C4×SD32 in GAP, Magma, Sage, TeX

C_4\times {\rm SD}_{32}
% in TeX

G:=Group("C4xSD32");
// GroupNames label

G:=SmallGroup(128,905);
// by ID

G=gap.SmallGroup(128,905);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,456,100,1123,570,360,4037,2028,124]);
// Polycyclic

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

׿
×
𝔽