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G = C4xSD32order 128 = 27

Direct product of C4 and SD32

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

Aliases: C4xSD32, C42.330D4, C16:8(C2xC4), (C4xC16):12C2, Q16:2(C2xC4), (C4xQ16):1C2, (C4xD8).4C2, D8.2(C2xC4), C2.14(C4xD8), C4.26(C4xD4), C4o2(C16:4C4), C16:4C4:14C2, (C2xC8).232D4, (C2xC4).173D8, C4o3(C2.D16), C2.4(C2xSD32), C2.D16.8C2, C2.4(C4oD16), C8.39(C4oD4), C4.12(C4oD8), C8.36(C22xC4), (C2xSD32).5C2, C22.62(C2xD8), C4o2(C2.Q32), C2.Q32:21C2, (C2xC16).70C22, (C2xC8).503C23, (C4xC8).397C22, (C2xD8).104C22, C2.D8.151C22, (C2xQ16).102C22, (C2xC4).769(C2xD4), SmallGroup(128,905)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C4xSD32
C1C2C4C2xC4C2xC8C4xC8C4xQ16 — C4xSD32
C1C2C4C8 — C4xSD32
C1C2xC4C42C4xC8 — C4xSD32
C1C2C2C2C2C4C4C2xC8 — C4xSD32

Generators and relations for C4xSD32
 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, C2, C4, C4, C4, C22, C22, C8, C8, C2xC4, C2xC4, D4, Q8, C23, C16, C16, C42, C42, C22:C4, C4:C4, C2xC8, D8, D8, Q16, Q16, C22xC4, C2xD4, C2xQ8, C4xC8, D4:C4, Q8:C4, C2.D8, C2xC16, SD32, C4xD4, C4xQ8, C2xD8, C2xQ16, C4xC16, C2.D16, C2.Q32, C16:4C4, C4xD8, C4xQ16, C2xSD32, C4xSD32
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, D8, C22xC4, C2xD4, C4oD4, SD32, C4xD4, C2xD8, C4oD8, C4xD8, C2xSD32, C4oD16, C4xSD32

Smallest permutation representation of C4xSD32
On 64 points
Generators in S64
(1 47 55 26)(2 48 56 27)(3 33 57 28)(4 34 58 29)(5 35 59 30)(6 36 60 31)(7 37 61 32)(8 38 62 17)(9 39 63 18)(10 40 64 19)(11 41 49 20)(12 42 50 21)(13 43 51 22)(14 44 52 23)(15 45 53 24)(16 46 54 25)
(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 55)(2 62)(3 53)(4 60)(5 51)(6 58)(7 49)(8 56)(9 63)(10 54)(11 61)(12 52)(13 59)(14 50)(15 57)(16 64)(17 48)(18 39)(19 46)(20 37)(21 44)(22 35)(23 42)(24 33)(25 40)(26 47)(27 38)(28 45)(29 36)(30 43)(31 34)(32 41)

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim1111111112222222
type+++++++++++
imageC1C2C2C2C2C2C2C2C4D4D4C4oD4D8SD32C4oD8C4oD16
kernelC4xSD32C4xC16C2.D16C2.Q32C16:4C4C4xD8C4xQ16C2xSD32SD32C42C2xC8C8C2xC4C4C4C2
# reps1111111181124848

Matrix representation of C4xSD32 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] >;

C4xSD32 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

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