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G = C4×C8○D4order 128 = 27

Direct product of C4 and C8○D4

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

Aliases: C4×C8○D4, D4.4C42, Q8.4C42, C42.589C23, D4(C4×C8), C8(C4×D4), C8(C4×Q8), Q8(C4×C8), C82(C8○D4), C8(C4×M4(2)), M4(2)(C4×C8), (C4×D4).41C4, (C4×Q8).38C4, C4.12(C2×C42), C8.48(C22×C4), C4.60(C23×C4), M4(2)⋊24(C2×C4), (C4×M4(2))⋊44C2, (C2×C8).638C23, (C2×C4).624C24, C42.280(C2×C4), (C4×C8).435C22, C22.2(C2×C42), C42(C82M4(2)), C82(C82M4(2)), C82M4(2)⋊39C2, C22.35(C23×C4), C2.16(C22×C42), C8⋊C4.172C22, C42(C82M4(2)), C23.136(C22×C4), (C22×C8).584C22, (C2×C42).1101C22, (C22×C4).1489C23, C42⋊C2.347C22, (C2×M4(2)).381C22, (C2×C4×C8)⋊41C2, C8(C4×C4○D4), C4○D4(C4×C8), C8(C2×C8○D4), (C4×C8)(C4×D4), (C4×C8)(C4×Q8), (C2×Q8)(C4×C8), (C2×C8)⋊33(C2×C4), (C2×C8)(C8○D4), C2.2(C2×C8○D4), C42(C2×C8○D4), (C4×C8)(C4×M4(2)), (C4×C8)2(C8⋊C4), C4⋊C4.244(C2×C4), (C4×C4○D4).32C2, C4○D4.36(C2×C4), (C2×C8○D4).23C2, (C2×C8)2(C4×M4(2)), (C4×C8)2(C2×M4(2)), (C4×C8)(C42⋊C2), (C2×D4).245(C2×C4), C22⋊C4.87(C2×C4), (C2×Q8).221(C2×C4), (C4×C8)(C82M4(2)), (C22×C4).382(C2×C4), (C2×C4).454(C22×C4), (C2×C4○D4).337C22, (C2×C8)(C4×C4○D4), (C4×C8)(C2×C4○D4), (C4×C8)(C2×C8○D4), SmallGroup(128,1606)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C4×C8○D4
C1C2C22C2×C4C22×C4C2×C4○D4C4×C4○D4 — C4×C8○D4
C1C2 — C4×C8○D4
C1C4×C8 — C4×C8○D4
C1C2C2C2×C4 — C4×C8○D4

Generators and relations for C4×C8○D4
 G = < a,b,c,d | a4=b8=d2=1, c2=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >

Subgroups: 316 in 286 conjugacy classes, 256 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×12], C4 [×6], C22, C22 [×6], C22 [×6], C8 [×16], C2×C4, C2×C4 [×23], C2×C4 [×12], D4 [×12], Q8 [×4], C23 [×3], C42, C42 [×9], C22⋊C4 [×6], C4⋊C4 [×6], C2×C8 [×32], M4(2) [×24], C22×C4 [×9], C2×D4 [×3], C2×Q8, C4○D4 [×8], C4×C8, C4×C8 [×9], C8⋊C4 [×6], C2×C42 [×3], C42⋊C2 [×3], C4×D4 [×6], C4×Q8 [×2], C22×C8 [×6], C2×M4(2) [×6], C8○D4 [×16], C2×C4○D4, C2×C4×C8 [×3], C4×M4(2) [×3], C82M4(2) [×6], C4×C4○D4, C2×C8○D4 [×2], C4×C8○D4
Quotients: C1, C2 [×15], C4 [×24], C22 [×35], C2×C4 [×84], C23 [×15], C42 [×16], C22×C4 [×42], C24, C2×C42 [×12], C8○D4 [×4], C23×C4 [×3], C22×C42, C2×C8○D4 [×2], C4×C8○D4

Smallest permutation representation of C4×C8○D4
On 64 points
Generators in S64
(1 39 63 47)(2 40 64 48)(3 33 57 41)(4 34 58 42)(5 35 59 43)(6 36 60 44)(7 37 61 45)(8 38 62 46)(9 18 30 49)(10 19 31 50)(11 20 32 51)(12 21 25 52)(13 22 26 53)(14 23 27 54)(15 24 28 55)(16 17 29 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 61 5 57)(2 62 6 58)(3 63 7 59)(4 64 8 60)(9 32 13 28)(10 25 14 29)(11 26 15 30)(12 27 16 31)(17 50 21 54)(18 51 22 55)(19 52 23 56)(20 53 24 49)(33 47 37 43)(34 48 38 44)(35 41 39 45)(36 42 40 46)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 17)(7 18)(8 19)(9 45)(10 46)(11 47)(12 48)(13 41)(14 42)(15 43)(16 44)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)(49 61)(50 62)(51 63)(52 64)(53 57)(54 58)(55 59)(56 60)

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

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

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

80 conjugacy classes

class 1 2A2B2C2D···2I4A···4L4M···4AD8A···8P8Q···8AN
order12222···24···44···48···88···8
size11112···21···12···21···12···2

80 irreducible representations

dim1111111112
type++++++
imageC1C2C2C2C2C2C4C4C4C8○D4
kernelC4×C8○D4C2×C4×C8C4×M4(2)C82M4(2)C4×C4○D4C2×C8○D4C4×D4C4×Q8C8○D4C4
# reps1336121243216

Matrix representation of C4×C8○D4 in GL3(𝔽17) generated by

1300
010
001
,
100
020
002
,
100
0130
004
,
100
004
0130
G:=sub<GL(3,GF(17))| [13,0,0,0,1,0,0,0,1],[1,0,0,0,2,0,0,0,2],[1,0,0,0,13,0,0,0,4],[1,0,0,0,0,13,0,4,0] >;

C4×C8○D4 in GAP, Magma, Sage, TeX

C_4\times C_8\circ D_4
% in TeX

G:=Group("C4xC8oD4");
// GroupNames label

G:=SmallGroup(128,1606);
// by ID

G=gap.SmallGroup(128,1606);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,232,521,172]);
// Polycyclic

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

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