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## G = (C2×D8)⋊10C4order 128 = 27

### 6th semidirect product of C2×D8 and C4 acting via C4/C2=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — (C2×D8)⋊10C4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C8 — C2×C8⋊C4 — (C2×D8)⋊10C4
 Lower central C1 — C2 — C2×C4 — (C2×D8)⋊10C4
 Upper central C1 — C23 — C2×C42 — (C2×D8)⋊10C4
 Jennings C1 — C2 — C2 — C22×C4 — (C2×D8)⋊10C4

Generators and relations for (C2×D8)⋊10C4
G = < a,b,c,d | a2=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=ab2c >

Subgroups: 484 in 188 conjugacy classes, 64 normal (24 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×4], C4 [×4], C22 [×3], C22 [×4], C22 [×20], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×12], D4 [×12], C23, C23 [×16], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×2], D8 [×8], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×4], C2×D4 [×10], C24 [×2], C8⋊C4 [×2], D4⋊C4 [×4], C4.Q8 [×2], C2×C42, C2×C22⋊C4 [×4], C2×C4⋊C4 [×2], C22×C8 [×2], C2×D8 [×4], C2×D8 [×4], C22×D4 [×2], C24.3C22 [×2], C2×C8⋊C4, C2×D4⋊C4 [×2], C2×C4.Q8, C22×D8, (C2×D8)⋊10C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C8⋊C22 [×4], C24.3C22, D8⋊C4 [×2], C82D4 [×2], C83D4 [×2], (C2×D8)⋊10C4

Smallest permutation representation of (C2×D8)⋊10C4
On 64 points
Generators in S64
(1 21)(2 22)(3 23)(4 24)(5 17)(6 18)(7 19)(8 20)(9 45)(10 46)(11 47)(12 48)(13 41)(14 42)(15 43)(16 44)(25 63)(26 64)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)
(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 20)(2 19)(3 18)(4 17)(5 24)(6 23)(7 22)(8 21)(9 46)(10 45)(11 44)(12 43)(13 42)(14 41)(15 48)(16 47)(25 30)(26 29)(27 28)(31 32)(33 36)(34 35)(37 40)(38 39)(49 52)(50 51)(53 56)(54 55)(57 58)(59 64)(60 63)(61 62)
(1 35 46 58)(2 38 47 61)(3 33 48 64)(4 36 41 59)(5 39 42 62)(6 34 43 57)(7 37 44 60)(8 40 45 63)(9 25 20 56)(10 28 21 51)(11 31 22 54)(12 26 23 49)(13 29 24 52)(14 32 17 55)(15 27 18 50)(16 30 19 53)

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

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

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

32 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 8A ··· 8H order 1 2 ··· 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 8 8 8 8 2 2 2 2 4 4 4 4 8 8 8 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 D4 D4 C4○D4 C8⋊C22 kernel (C2×D8)⋊10C4 C24.3C22 C2×C8⋊C4 C2×D4⋊C4 C2×C4.Q8 C22×D8 C2×D8 C2×C8 C22×C4 C2×C4 C22 # reps 1 2 1 2 1 1 8 6 2 4 4

Matrix representation of (C2×D8)⋊10C4 in GL8(𝔽17)

 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16
,
 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 1 0 0 2 0 0 0 0 1 0 1 1 0 0 0 0 16 0 0 16 0 0 0 0 16 1 0 16
,
 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 16 0 0 15 0 0 0 0 0 0 1 16 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1
,
 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 5 8 9 9 0 0 0 0 13 4 0 9 0 0 0 0 12 9 8 4 0 0 0 0 4 0 4 0

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,16,16,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,2,1,16,16],[0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,16,1,1],[0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,5,13,12,4,0,0,0,0,8,4,9,0,0,0,0,0,9,0,8,4,0,0,0,0,9,9,4,0] >;

(C2×D8)⋊10C4 in GAP, Magma, Sage, TeX

(C_2\times D_8)\rtimes_{10}C_4
% in TeX

G:=Group("(C2xD8):10C4");
// GroupNames label

G:=SmallGroup(128,704);
// by ID

G=gap.SmallGroup(128,704);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,723,100,2019,248,2804,172]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=a*b^2*c>;
// generators/relations

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