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## G = C4×C4.4D4order 128 = 27

### Direct product of C4 and C4.4D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C4×C4.4D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C43 — C4×C4.4D4
 Lower central C1 — C22 — C4×C4.4D4
 Upper central C1 — C22×C4 — C4×C4.4D4
 Jennings C1 — C23 — C4×C4.4D4

Generators and relations for C4×C4.4D4
G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 540 in 326 conjugacy classes, 160 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×8], C4 [×18], C22 [×3], C22 [×4], C22 [×20], C2×C4 [×26], C2×C4 [×34], D4 [×8], Q8 [×8], C23, C23 [×4], C23 [×12], C42 [×8], C42 [×14], C22⋊C4 [×16], C22⋊C4 [×12], C4⋊C4 [×8], C22×C4 [×3], C22×C4 [×10], C22×C4 [×12], C2×D4 [×4], C2×D4 [×4], C2×Q8 [×4], C2×Q8 [×4], C24 [×2], C2.C42 [×8], C2×C42 [×3], C2×C42 [×6], C2×C22⋊C4 [×10], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C4×D4 [×4], C4×Q8 [×4], C4.4D4 [×8], C23×C4 [×2], C22×D4, C22×Q8, C43, C4×C22⋊C4 [×4], C428C4, C24.C22 [×4], C24.3C22, C23.67C23, C2×C4×D4, C2×C4×Q8, C2×C4.4D4, C4×C4.4D4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×10], C24, C4×D4 [×4], C4.4D4 [×4], C23×C4, C22×D4, C2×C4○D4 [×5], C2×C4×D4, C4×C4○D4 [×2], C2×C4.4D4, C23.36C23 [×2], C22.26C24, C4×C4.4D4

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

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4AF 4AG ··· 4AR order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 4 ··· 4 size 1 1 ··· 1 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C4 D4 C4○D4 kernel C4×C4.4D4 C43 C4×C22⋊C4 C42⋊8C4 C24.C22 C24.3C22 C23.67C23 C2×C4×D4 C2×C4×Q8 C2×C4.4D4 C4.4D4 C42 C2×C4 # reps 1 1 4 1 4 1 1 1 1 1 16 4 20

Matrix representation of C4×C4.4D4 in GL5(𝔽5)

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

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

C4×C4.4D4 in GAP, Magma, Sage, TeX

C_4\times C_4._4D_4
% in TeX

G:=Group("C4xC4.4D4");
// GroupNames label

G:=SmallGroup(128,1035);
// by ID

G=gap.SmallGroup(128,1035);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,100,248]);
// Polycyclic

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

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