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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 [×3], C2 [×2], C4 [×6], C4 [×9], C22 [×3], C22 [×2], C8 [×8], C2×C4 [×2], C2×C4 [×12], C2×C4 [×11], Q8 [×8], C23, C42 [×4], C42 [×4], C4⋊C4 [×6], C2×C8 [×4], M4(2) [×8], M4(2) [×4], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×4], C4×C8 [×2], C8⋊C4 [×2], C4.10D4 [×8], C2×C42, C2×C42 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C4×Q8 [×4], C2×M4(2) [×4], C22×Q8, C22.C42 [×2], C4×M4(2) [×2], C2×C4.10D4 [×2], C2×C4×Q8, C4×C4.10D4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×4], C23, C42 [×4], C22⋊C4 [×4], C22×C4 [×3], C2×D4 [×2], C4○D4 [×2], C4.10D4 [×2], C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4 [×4], 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 15 47 21)(2 16 48 22)(3 9 41 23)(4 10 42 24)(5 11 43 17)(6 12 44 18)(7 13 45 19)(8 14 46 20)(25 64 51 34)(26 57 52 35)(27 58 53 36)(28 59 54 37)(29 60 55 38)(30 61 56 39)(31 62 49 40)(32 63 50 33)
(1 41 5 45)(2 46 6 42)(3 43 7 47)(4 48 8 44)(9 17 13 21)(10 22 14 18)(11 19 15 23)(12 24 16 20)(25 49 29 53)(26 54 30 50)(27 51 31 55)(28 56 32 52)(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 31 21 51 13 27 17 55)(10 50 18 30 14 54 22 26)(11 29 23 49 15 25 19 53)(12 56 20 28 16 52 24 32)

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

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

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

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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