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G = C2×SD16⋊C4  order 128 = 27

Direct product of C2 and SD16⋊C4

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

Aliases: C2×SD16⋊C4, C42.202D4, C42.269C23, C8⋊3(C22×C4), C4.45(C4×D4), (C2×SD16)⋊9C4, SD16⋊6(C2×C4), Q8⋊2(C22×C4), C4.17(C23×C4), (C4×Q8)⋊75C22, D4.3(C22×C4), C8⋊C4⋊34C22, C2.D8⋊64C22, C4⋊C4.357C23, (C2×C4).197C24, (C2×C8).408C23, C23.843(C2×D4), C22.118(C4×D4), (C22×C4).707D4, Q8⋊C4⋊90C22, (C4×D4).290C22, (C2×D4).367C23, (C2×Q8).340C23, (C22×SD16).3C2, (C2×C42).762C22, (C22×C8).436C22, C22.141(C22×D4), D4⋊C4.194C22, C22.107(C8⋊C22), (C22×C4).1513C23, (C2×SD16).105C22, (C22×D4).557C22, C22.96(C8.C22), (C22×Q8).461C22, (C2×C4×Q8)⋊32C2, C2.57(C2×C4×D4), (C2×C8)⋊11(C2×C4), (C2×C8⋊C4)⋊4C2, (C2×C4×D4).74C2, C4.5(C2×C4○D4), (C2×Q8)⋊28(C2×C4), (C2×C2.D8)⋊38C2, C2.4(C2×C8⋊C22), C2.4(C2×C8.C22), (C2×Q8⋊C4)⋊52C2, (C2×D4).176(C2×C4), (C2×C4).1209(C2×D4), (C2×D4⋊C4).37C2, (C2×C4).689(C4○D4), (C2×C4⋊C4).909C22, (C2×C4).468(C22×C4), SmallGroup(128,1672)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C4 — C2×SD16⋊C4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C2×C4×D4 — C2×SD16⋊C4
Lower central
C1 — C2 — C4 — C2×SD16⋊C4
Upper central
C1 — C23 — C2×C42 — C2×SD16⋊C4
Jennings
C1 — C2 — C2 — C2×C4 — C2×SD16⋊C4

Subgroups: 476 in 272 conjugacy classes, 148 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×12], C22, C22 [×6], C22 [×16], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×26], D4 [×4], D4 [×6], Q8 [×4], Q8 [×6], C23, C23 [×10], C42 [×4], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×8], C2×C8 [×2], SD16 [×16], C22×C4 [×3], C22×C4 [×11], C2×D4 [×6], C2×D4 [×3], C2×Q8 [×6], C2×Q8 [×3], C24, C8⋊C4 [×4], D4⋊C4 [×4], Q8⋊C4 [×4], C2.D8 [×4], C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4, C4×D4 [×4], C4×D4 [×2], C4×Q8 [×4], C4×Q8 [×2], C22×C8 [×2], C2×SD16 [×12], C23×C4, C22×D4, C22×Q8, C2×C8⋊C4, C2×D4⋊C4, C2×Q8⋊C4, C2×C2.D8, SD16⋊C4 [×8], C2×C4×D4, C2×C4×Q8, C22×SD16, C2×SD16⋊C4

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C8⋊C22 [×2], C8.C22 [×2], C23×C4, C22×D4, C2×C4○D4, SD16⋊C4 [×4], C2×C4×D4, C2×C8⋊C22, C2×C8.C22, C2×SD16⋊C4

Generators and relations
 G = < a,b,c,d | a2=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd-1=b5, cd=dc >

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

160000000
016000000
00100000
00010000
00001000
00000100
00000010
00000001
,
01000000
160000000
00040000
00400000
0000215215
00002222
0000215152
0000221515
,
10000000
016000000
001600000
00010000
000016000
00000100
000000160
00000001
,
160000000
016000000
001300000
000130000
00000010
00000001
000016000
000001600

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,2,2,2,2,0,0,0,0,15,2,15,2,0,0,0,0,2,2,15,15,0,0,0,0,15,2,2,15],[1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
 

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4L4M···4X8A···8H
order12···222224···44···48···8
size11···144442···24···44···4

44 irreducible representations

dim111111111122244
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C4D4D4C4○D4C8⋊C22C8.C22
kernelC2×SD16⋊C4C2×C8⋊C4C2×D4⋊C4C2×Q8⋊C4C2×C2.D8SD16⋊C4C2×C4×D4C2×C4×Q8C22×SD16C2×SD16C42C22×C4C2×C4C22C22
# reps1111181111622422

In GAP, Magma, Sage, TeX 

C_2\times SD_{16}\rtimes C_4
 
% in TeX
 
G:=Group("C2xSD16:C4");
 
// GroupNames label
 
G:=SmallGroup(128,1672);
 
// by ID
 
G=gap.SmallGroup(128,1672);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,184,2804,1411,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d^-1=b^5,c*d=d*c>;
 
// generators/relations
 

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