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## G = C4×C4○D8order 128 = 27

### Direct product of C4 and C4○D8

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C4×C4○D8
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c3 >

Subgroups: 396 in 250 conjugacy classes, 144 normal (22 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, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, D8, SD16, Q16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C2×C4×C8, C23.24D4, C23.25D4, C4×D8, C4×SD16, C4×Q16, C4×C4○D4, C2×C4○D8, C4×C4○D8
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C4○D8, C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, C2×C4○D8, C4×C4○D8

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

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A ··· 4L 4M ··· 4R 4S ··· 4AD 8A ··· 8P order 1 2 2 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 2 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C4 D4 D4 C4○D4 C4○D8 kernel C4×C4○D8 C2×C4×C8 C23.24D4 C23.25D4 C4×D8 C4×SD16 C4×Q16 C4×C4○D4 C2×C4○D8 C4○D8 C42 C22×C4 C2×C4 C4 # reps 1 1 2 1 2 4 2 2 1 16 2 2 4 16

Matrix representation of C4×C4○D8 in GL3(𝔽17) generated by

 4 0 0 0 1 0 0 0 1
,
 1 0 0 0 13 0 0 0 13
,
 1 0 0 0 3 14 0 3 3
,
 1 0 0 0 14 3 0 3 3
G:=sub<GL(3,GF(17))| [4,0,0,0,1,0,0,0,1],[1,0,0,0,13,0,0,0,13],[1,0,0,0,3,3,0,14,3],[1,0,0,0,14,3,0,3,3] >;

C4×C4○D8 in GAP, Magma, Sage, TeX

C_4\times C_4\circ D_8
% in TeX

G:=Group("C4xC4oD8");
// GroupNames label

G:=SmallGroup(128,1671);
// by ID

G=gap.SmallGroup(128,1671);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,248,2804,1411,172]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^3>;
// generators/relations

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