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G = C2×D8⋊C4order 128 = 27

Direct product of C2 and D8⋊C4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×D8⋊C4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C2×C4×D4 — C2×D8⋊C4
 Lower central C1 — C2 — C4 — C2×D8⋊C4
 Upper central C1 — C23 — C2×C42 — C2×D8⋊C4
 Jennings C1 — C2 — C2 — C2×C4 — C2×D8⋊C4

Generators and relations for C2×D8⋊C4
G = < a,b,c,d | a2=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b5, dcd-1=b4c >

Subgroups: 604 in 304 conjugacy classes, 148 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C8⋊C4, D4⋊C4, C4.Q8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C22×C8, C2×D8, C23×C4, C22×D4, C2×C8⋊C4, C2×D4⋊C4, C2×C4.Q8, D8⋊C4, C2×C4×D4, C22×D8, C2×D8⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C8⋊C22, C23×C4, C22×D4, C2×C4○D4, D8⋊C4, C2×C4×D4, C2×C8⋊C22, C2×D8⋊C4

Smallest permutation representation of C2×D8⋊C4
On 64 points
Generators in S64
(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(9 63)(10 64)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 49)(48 50)
(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 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 59)(10 58)(11 57)(12 64)(13 63)(14 62)(15 61)(16 60)(17 35)(18 34)(19 33)(20 40)(21 39)(22 38)(23 37)(24 36)(41 51)(42 50)(43 49)(44 56)(45 55)(46 54)(47 53)(48 52)
(1 38 49 15)(2 35 50 12)(3 40 51 9)(4 37 52 14)(5 34 53 11)(6 39 54 16)(7 36 55 13)(8 33 56 10)(17 46 64 27)(18 43 57 32)(19 48 58 29)(20 45 59 26)(21 42 60 31)(22 47 61 28)(23 44 62 25)(24 41 63 30)

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A ··· 4L 4M ··· 4T 8A ··· 8H order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

44 irreducible representations

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

Matrix representation of C2×D8⋊C4 in GL8(𝔽17)

 16 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 16 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 16 0 0 0 0 0 0 0 0 16
,
 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 16 15 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 15 2 15 0 0 0 0 1 15 1 0 0 0 0 0 1 16 0 2 0 0 0 0 9 0 16 2
,
 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 1 1 0 0 0 0 0 0 0 0 1 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 16 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 1 15 2 0 0 0 0 0 1 16 0 2 0 0 0 0 1 0 16 2 0 0 0 0 0 1 16 1

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,0,1,1,9,0,0,0,0,15,15,16,0,0,0,0,0,2,1,0,16,0,0,0,0,15,0,2,2],[16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,16,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,15,16,0,1,0,0,0,0,2,0,16,16,0,0,0,0,0,2,2,1] >;

C2×D8⋊C4 in GAP, Magma, Sage, TeX

C_2\times D_8\rtimes C_4
% in TeX

G:=Group("C2xD8:C4");
// GroupNames label

G:=SmallGroup(128,1674);
// by ID

G=gap.SmallGroup(128,1674);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,520,2804,1411,172]);
// 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^-1,d*b*d^-1=b^5,d*c*d^-1=b^4*c>;
// generators/relations

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