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## G = D4×C4⋊C4order 128 = 27

### Direct product of D4 and C4⋊C4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — D4×C4⋊C4
 Chief series C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C4×D4 — D4×C4⋊C4
 Lower central C1 — C22 — D4×C4⋊C4
 Upper central C1 — C23 — D4×C4⋊C4
 Jennings C1 — C23 — D4×C4⋊C4

Generators and relations for D4×C4⋊C4
G = < a,b,c,d | a4=b2=c4=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 636 in 388 conjugacy classes, 196 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C23×C4, C22×D4, C4×C4⋊C4, C23.7Q8, C429C4, C23.8Q8, C23.65C23, C22×C4⋊C4, C2×C4×D4, C2×C4×D4, D4×C4⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C2×C4⋊C4, C4×D4, C23×C4, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C22×C4⋊C4, C2×C4×D4, C23.33C23, D42, D46D4, D4×Q8, D43Q8, D4×C4⋊C4

Smallest permutation representation of D4×C4⋊C4
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 50)(2 49)(3 52)(4 51)(5 56)(6 55)(7 54)(8 53)(9 35)(10 34)(11 33)(12 36)(13 39)(14 38)(15 37)(16 40)(17 44)(18 43)(19 42)(20 41)(21 48)(22 47)(23 46)(24 45)(25 29)(26 32)(27 31)(28 30)(57 63)(58 62)(59 61)(60 64)
(1 13 51 40)(2 14 52 37)(3 15 49 38)(4 16 50 39)(5 44 55 20)(6 41 56 17)(7 42 53 18)(8 43 54 19)(9 31 36 28)(10 32 33 25)(11 29 34 26)(12 30 35 27)(21 61 45 60)(22 62 46 57)(23 63 47 58)(24 64 48 59)
(1 56 29 63)(2 53 30 64)(3 54 31 61)(4 55 32 62)(5 25 57 50)(6 26 58 51)(7 27 59 52)(8 28 60 49)(9 45 38 19)(10 46 39 20)(11 47 40 17)(12 48 37 18)(13 41 34 23)(14 42 35 24)(15 43 36 21)(16 44 33 22)

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

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

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

50 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A ··· 4P 4Q ··· 4AH order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 D4 Q8 C4○D4 2+ 1+4 2- 1+4 kernel D4×C4⋊C4 C4×C4⋊C4 C23.7Q8 C42⋊9C4 C23.8Q8 C23.65C23 C22×C4⋊C4 C2×C4×D4 C4×D4 C4⋊C4 C2×D4 C2×D4 C2×C4 C22 C22 # reps 1 1 2 1 4 2 2 3 16 4 4 4 4 1 1

Matrix representation of D4×C4⋊C4 in GL5(𝔽5)

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

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

D4×C4⋊C4 in GAP, Magma, Sage, TeX

D_4\times C_4\rtimes C_4
% in TeX

G:=Group("D4xC4:C4");
// GroupNames label

G:=SmallGroup(128,1080);
// by ID

G=gap.SmallGroup(128,1080);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,184,346]);
// Polycyclic

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

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