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## G = C4×D4⋊C4order 128 = 27

### Direct product of C4 and D4⋊C4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C4×D4⋊C4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C2×C4×C8 — C4×D4⋊C4
 Lower central C1 — C2 — C4 — C4×D4⋊C4
 Upper central C1 — C22×C4 — C2×C42 — C4×D4⋊C4
 Jennings C1 — C2 — C2 — C22×C4 — C4×D4⋊C4

Generators and relations for C4×D4⋊C4
G = < a,b,c,d | a4=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >

Subgroups: 396 in 198 conjugacy classes, 92 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×6], C4 [×8], C22 [×3], C22 [×4], C22 [×16], C8 [×4], C2×C4 [×2], C2×C4 [×12], C2×C4 [×22], D4 [×4], D4 [×6], C23, C23 [×10], C42 [×4], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×6], C4⋊C4 [×3], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×11], C2×D4 [×6], C2×D4 [×3], C24, C2.C42, C4×C8 [×2], D4⋊C4 [×8], C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4 [×2], C4×D4 [×4], C4×D4 [×2], C22×C8 [×2], C23×C4, C22×D4, C22.4Q16 [×2], C4×C4⋊C4, C2×C4×C8, C2×D4⋊C4 [×2], C2×C4×D4, C4×D4⋊C4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×4], C23, C42 [×4], C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4 [×3], C2×D4 [×2], C4○D4 [×2], D4⋊C4 [×4], C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4 [×4], C2×D8, C2×SD16, C4○D8 [×2], C4×C22⋊C4, C2×D4⋊C4, C23.24D4, C4×D8 [×2], C4×SD16 [×2], C4×D4⋊C4

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

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 D4 D4 D8 SD16 C4○D4 C4○D8 kernel C4×D4⋊C4 C22.4Q16 C4×C4⋊C4 C2×C4×C8 C2×D4⋊C4 C2×C4×D4 D4⋊C4 C4×D4 C42 C22×C4 C2×C4 C2×C4 C2×C4 C22 # reps 1 2 1 1 2 1 16 8 2 2 4 4 4 8

Matrix representation of C4×D4⋊C4 in GL4(𝔽17) generated by

 13 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 1 0 0 0 0 1 0 0 0 0 0 16 0 0 1 0
,
 16 0 0 0 0 1 0 0 0 0 1 0 0 0 0 16
,
 16 0 0 0 0 13 0 0 0 0 3 14 0 0 14 14
G:=sub<GL(4,GF(17))| [13,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,16,0],[16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16],[16,0,0,0,0,13,0,0,0,0,3,14,0,0,14,14] >;

C4×D4⋊C4 in GAP, Magma, Sage, TeX

C_4\times D_4\rtimes C_4
% in TeX

G:=Group("C4xD4:C4");
// GroupNames label

G:=SmallGroup(128,492);
// by ID

G=gap.SmallGroup(128,492);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,100,2019,248,4037,124]);
// Polycyclic

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

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