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## G = C2×SD16⋊C4order 128 = 27

### Direct product of C2 and SD16⋊C4

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

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

Generators and relations for C2×SD16⋊C4
G = < a,b,c,d | a2=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd-1=b5, cd=dc >

Subgroups: 476 in 272 conjugacy classes, 148 normal (28 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C4×Q8, C22×C8, C2×SD16, C23×C4, C22×D4, C22×Q8, C2×C8⋊C4, C2×D4⋊C4, C2×Q8⋊C4, C2×C2.D8, SD16⋊C4, C2×C4×D4, C2×C4×Q8, C22×SD16, C2×SD16⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C8⋊C22, C8.C22, C23×C4, C22×D4, C2×C4○D4, SD16⋊C4, C2×C4×D4, C2×C8⋊C22, C2×C8.C22, C2×SD16⋊C4

Smallest permutation representation of C2×SD16⋊C4
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)]])

44 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4L 4M ··· 4X 8A ··· 8H order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 4 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C4 D4 D4 C4○D4 C8⋊C22 C8.C22 kernel C2×SD16⋊C4 C2×C8⋊C4 C2×D4⋊C4 C2×Q8⋊C4 C2×C2.D8 SD16⋊C4 C2×C4×D4 C2×C4×Q8 C22×SD16 C2×SD16 C42 C22×C4 C2×C4 C22 C22 # reps 1 1 1 1 1 8 1 1 1 16 2 2 4 2 2

Matrix representation of C2×SD16⋊C4 in GL8(𝔽17)

 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 1 0 0 0 0 0 0 16 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 15 2 15 0 0 0 0 2 2 2 2 0 0 0 0 2 15 15 2 0 0 0 0 2 2 15 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 1 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0

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] >;

C2×SD16⋊C4 in GAP, Magma, Sage, TeX

C_2\times {\rm 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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