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## G = D4×C42order 128 = 27

### Direct product of C42 and D4

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for D4×C42
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 636 in 462 conjugacy classes, 288 normal (9 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, C24, C2.C42, C2×C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C22×D4, C43, C4×C22⋊C4, C4×C4⋊C4, C22×C42, C2×C4×D4, D4×C42
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C42, C22×C4, C2×D4, C4○D4, C24, C2×C42, C4×D4, C23×C4, C22×D4, C2×C4○D4, C22×C42, C2×C4×D4, C4×C4○D4, D4×C42

Smallest permutation representation of D4×C42
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 31 27 6)(2 32 28 7)(3 29 25 8)(4 30 26 5)(9 33 13 21)(10 34 14 22)(11 35 15 23)(12 36 16 24)(17 47 38 43)(18 48 39 44)(19 45 40 41)(20 46 37 42)(49 55 63 59)(50 56 64 60)(51 53 61 57)(52 54 62 58)
(1 58 11 46)(2 59 12 47)(3 60 9 48)(4 57 10 45)(5 61 22 19)(6 62 23 20)(7 63 24 17)(8 64 21 18)(13 44 25 56)(14 41 26 53)(15 42 27 54)(16 43 28 55)(29 50 33 39)(30 51 34 40)(31 52 35 37)(32 49 36 38)
(1 25)(2 26)(3 27)(4 28)(5 32)(6 29)(7 30)(8 31)(9 15)(10 16)(11 13)(12 14)(17 51)(18 52)(19 49)(20 50)(21 35)(22 36)(23 33)(24 34)(37 64)(38 61)(39 62)(40 63)(41 59)(42 60)(43 57)(44 58)(45 55)(46 56)(47 53)(48 54)

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

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

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

80 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A ··· 4X 4Y ··· 4BL order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

80 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 D4×C42 C43 C4×C22⋊C4 C4×C4⋊C4 C22×C42 C2×C4×D4 C4×D4 C42 C2×C4 # reps 1 1 6 3 2 3 48 4 12

Matrix representation of D4×C42 in GL4(𝔽5) generated by

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

D4×C42 in GAP, Magma, Sage, TeX

D_4\times C_4^2
% in TeX

G:=Group("D4xC4^2");
// GroupNames label

G:=SmallGroup(128,1003);
// by ID

G=gap.SmallGroup(128,1003);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,352,136]);
// 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,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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