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## G = C4×C4⋊1D4order 128 = 27

### Direct product of C4 and C4⋊1D4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C4×C4⋊1D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C43 — C4×C4⋊1D4
 Lower central C1 — C22 — C4×C4⋊1D4
 Upper central C1 — C22×C4 — C4×C4⋊1D4
 Jennings C1 — C23 — C4×C4⋊1D4

Generators and relations for C4×C41D4
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 828 in 462 conjugacy classes, 184 normal (10 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C41D4, C23×C4, C22×D4, C43, C429C4, C24.3C22, C2×C4×D4, C2×C41D4, C4×C41D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C41D4, C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, C2×C41D4, C22.26C24, C4×C41D4

Smallest permutation representation of C4×C41D4
On 64 points
Generators in S64
(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 63 8 58)(2 64 5 59)(3 61 6 60)(4 62 7 57)(9 13 21 50)(10 14 22 51)(11 15 23 52)(12 16 24 49)(17 54 38 33)(18 55 39 34)(19 56 40 35)(20 53 37 36)(25 46 30 41)(26 47 31 42)(27 48 32 43)(28 45 29 44)
(1 28 23 39)(2 25 24 40)(3 26 21 37)(4 27 22 38)(5 30 12 19)(6 31 9 20)(7 32 10 17)(8 29 11 18)(13 53 60 42)(14 54 57 43)(15 55 58 44)(16 56 59 41)(33 62 48 51)(34 63 45 52)(35 64 46 49)(36 61 47 50)
(1 42)(2 43)(3 44)(4 41)(5 48)(6 45)(7 46)(8 47)(9 34)(10 35)(11 36)(12 33)(13 39)(14 40)(15 37)(16 38)(17 49)(18 50)(19 51)(20 52)(21 55)(22 56)(23 53)(24 54)(25 57)(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(32 64)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A ··· 4H 4I ··· 4AF 4AG ··· 4AN order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 4 ··· 4 size 1 1 ··· 1 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C2 C4 D4 C4○D4 kernel C4×C4⋊1D4 C43 C42⋊9C4 C24.3C22 C2×C4×D4 C2×C4⋊1D4 C4⋊1D4 C42 C2×C4 # reps 1 1 1 6 6 1 16 12 12

Matrix representation of C4×C41D4 in GL5(𝔽5)

 2 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 3 0 0 0 0 0 3
,
 4 0 0 0 0 0 0 1 0 0 0 4 0 0 0 0 0 0 4 4 0 0 0 2 1
,
 1 0 0 0 0 0 0 1 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 4
,
 4 0 0 0 0 0 1 0 0 0 0 0 4 0 0 0 0 0 4 4 0 0 0 0 1

G:=sub<GL(5,GF(5))| [2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,3],[4,0,0,0,0,0,0,4,0,0,0,1,0,0,0,0,0,0,4,2,0,0,0,4,1],[1,0,0,0,0,0,0,4,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,4,1] >;

C4×C41D4 in GAP, Magma, Sage, TeX

C_4\times C_4\rtimes_1D_4
% in TeX

G:=Group("C4xC4:1D4");
// GroupNames label

G:=SmallGroup(128,1038);
// by ID

G=gap.SmallGroup(128,1038);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,184,248]);
// Polycyclic

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

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