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## G = C2.(C4×D8)  order 128 = 27

### 3rd central stem extension by C2 of C4×D8

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2.(C4×D8)
G = < a,b,c,d | a2=b4=c8=d2=1, cbc-1=ab=ba, ac=ca, ad=da, bd=db, dcd=ac-1 >

Subgroups: 396 in 181 conjugacy classes, 66 normal (44 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, D4⋊C4, D4⋊C4, C2.D8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C22×C8, C23×C4, C22×D4, C22.7C42, C22.4Q16, C23.65C23, C2×D4⋊C4, C2×C2.D8, C2×C4×D4, C2.(C4×D8)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, D8, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C2×D8, C4○D8, C8⋊C22, C8.C22, C23.8Q8, C4×D8, SD16⋊C4, C22⋊D8, D4.7D4, D4⋊Q8, D4.Q8, C2.(C4×D8)

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

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4N 4O 4P 4Q 4R 8A ··· 8H order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 4 4 4 4 2 ··· 2 4 ··· 4 8 8 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 C4 D4 D4 D4 Q8 D8 C4○D4 C4○D8 C8⋊C22 C8.C22 kernel C2.(C4×D8) C22.7C42 C22.4Q16 C23.65C23 C2×D4⋊C4 C2×C2.D8 C2×C4×D4 D4⋊C4 C4⋊C4 C22×C4 C2×D4 C2×D4 C2×C4 C2×C4 C22 C22 C22 # reps 1 1 1 1 2 1 1 8 2 2 2 2 4 4 4 1 1

Matrix representation of C2.(C4×D8) in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 13 0 0 0 0 0 2 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 16 13 0 0 0 0 0 1 0 0 0 0 0 0 7 8 0 0 0 0 15 10 0 0 0 0 0 0 3 3 0 0 0 0 14 3
,
 1 0 0 0 0 0 8 16 0 0 0 0 0 0 1 0 0 0 0 0 11 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[13,2,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[16,0,0,0,0,0,13,1,0,0,0,0,0,0,7,15,0,0,0,0,8,10,0,0,0,0,0,0,3,14,0,0,0,0,3,3],[1,8,0,0,0,0,0,16,0,0,0,0,0,0,1,11,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;

C2.(C4×D8) in GAP, Magma, Sage, TeX

C_2.(C_4\times D_8)
% in TeX

G:=Group("C2.(C4xD8)");
// GroupNames label

G:=SmallGroup(128,594);
// by ID

G=gap.SmallGroup(128,594);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,2019,1018,521,248,2804,1411,718,172]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^8=d^2=1,c*b*c^-1=a*b=b*a,a*c=c*a,a*d=d*a,b*d=d*b,d*c*d=a*c^-1>;
// generators/relations

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