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## G = C4×C4.10D4order 128 = 27

### Direct product of C4 and C4.10D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C4×C4.10D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C4×M4(2) — C4×C4.10D4
 Lower central C1 — C2 — C22 — C4×C4.10D4
 Upper central C1 — C2×C4 — C2×C42 — C4×C4.10D4
 Jennings C1 — C2 — C2 — C22×C4 — C4×C4.10D4

Generators and relations for C4×C4.10D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=b-1c3 >

Subgroups: 228 in 148 conjugacy classes, 78 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×Q8, C2×Q8, C4×C8, C8⋊C4, C4.10D4, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C2×M4(2), C22×Q8, C22.C42, C4×M4(2), C2×C4.10D4, C2×C4×Q8, C4×C4.10D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C42, C22⋊C4, C22×C4, C2×D4, C4○D4, C4.10D4, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C4×C22⋊C4, C2×C4.10D4, M4(2).8C22, C4×C4.10D4

Smallest permutation representation of C4×C4.10D4
On 64 points
Generators in S64
(1 9 47 21)(2 10 48 22)(3 11 41 23)(4 12 42 24)(5 13 43 17)(6 14 44 18)(7 15 45 19)(8 16 46 20)(25 64 49 34)(26 57 50 35)(27 58 51 36)(28 59 52 37)(29 60 53 38)(30 61 54 39)(31 62 55 40)(32 63 56 33)
(1 41 5 45)(2 46 6 42)(3 43 7 47)(4 48 8 44)(9 23 13 19)(10 20 14 24)(11 17 15 21)(12 22 16 18)(25 55 29 51)(26 52 30 56)(27 49 31 53)(28 54 32 50)(33 57 37 61)(34 62 38 58)(35 59 39 63)(36 64 40 60)
(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 34 45 58 5 38 41 62)(2 57 42 33 6 61 46 37)(3 40 47 64 7 36 43 60)(4 63 44 39 8 59 48 35)(9 25 19 51 13 29 23 55)(10 50 24 32 14 54 20 28)(11 31 21 49 15 27 17 53)(12 56 18 30 16 52 22 26)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4N 4O ··· 4V 8A ··· 8P order 1 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 2 1 1 1 1 2 ··· 2 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + - image C1 C2 C2 C2 C2 C4 C4 C4 D4 C4○D4 C4.10D4 M4(2).8C22 kernel C4×C4.10D4 C22.C42 C4×M4(2) C2×C4.10D4 C2×C4×Q8 C4.10D4 C2×C42 C2×C4⋊C4 C42 C2×C4 C4 C2 # reps 1 2 2 2 1 16 4 4 4 4 2 2

Matrix representation of C4×C4.10D4 in GL6(𝔽17)

 4 0 0 0 0 0 0 4 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 13 0 0 0 0 0 2 4 0 0 0 0 0 0 4 0 0 0 0 0 15 13
,
 1 8 0 0 0 0 0 16 0 0 0 0 0 0 0 0 4 0 0 0 0 0 15 13 0 0 16 0 0 0 0 0 0 16 0 0
,
 5 14 0 0 0 0 3 12 0 0 0 0 0 0 0 0 11 10 0 0 0 0 15 6 0 0 10 6 0 0 0 0 3 7 0 0

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,2,0,0,0,0,0,4,0,0,0,0,0,0,4,15,0,0,0,0,0,13],[1,0,0,0,0,0,8,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,4,15,0,0,0,0,0,13,0,0],[5,3,0,0,0,0,14,12,0,0,0,0,0,0,0,0,10,3,0,0,0,0,6,7,0,0,11,15,0,0,0,0,10,6,0,0] >;

C4×C4.10D4 in GAP, Magma, Sage, TeX

C_4\times C_4._{10}D_4
% in TeX

G:=Group("C4xC4.10D4");
// GroupNames label

G:=SmallGroup(128,488);
// by ID

G=gap.SmallGroup(128,488);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,436,1018,4037,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=c*b*c^-1=b^-1,a*b=b*a,a*c=c*a,a*d=d*a,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations

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