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G = C4×C4○D8order 128 = 27

Direct product of C4 and C4○D8

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

Aliases: C4×C4○D8, C42D8, C42Q16, C42SD16, C42.461D4, C42.686C23, C42(C4×D8), C42(C4×Q16), C42(C2×D8), C42(C4×D8), (C4×D8)⋊47C2, D814(C2×C4), C42(C4×SD16), C42(C4×Q16), C42(C2×Q16), (C4×Q16)⋊47C2, Q1614(C2×C4), C4.185(C4×D4), C42(C4×SD16), C42(C2×SD16), C22.4(C4×D4), C42(C2.D8), C42(C4.Q8), (C4×SD16)⋊65C2, SD1610(C2×C4), C8.45(C22×C4), C4.16(C23×C4), D4.2(C22×C4), Q8.2(C22×C4), C42(D4⋊C4), C4⋊C4.356C23, (C2×C8).478C23, (C2×C4).196C24, (C4×C8).407C22, C42(Q8⋊C4), C23.380(C2×D4), (C22×C4).563D4, (C4×D4).289C22, (C2×D8).172C22, (C2×D4).366C23, (C4×Q8).272C22, (C2×Q8).339C23, C2.D8.234C22, C4.Q8.185C22, C42(C23.25D4), C42(C23.24D4), C23.24D444C2, C23.25D434C2, (C22×C8).513C22, (C2×Q16).168C22, C22.140(C22×D4), D4⋊C4.215C22, (C2×C42).1114C22, (C22×C4).1512C23, Q8⋊C4.215C22, (C2×SD16).177C22, C42(C23.24D4), C42(C23.25D4), C42⋊C2.295C22, (C2×C4×C8)⋊28C2, C2.56(C2×C4×D4), (C2×C8)⋊30(C2×C4), C4○D47(C2×C4), (C4×C4○D4)⋊3C2, C4.4(C2×C4○D4), C2.6(C2×C4○D8), C42(C2×C4○D8), (C2×C4○D8).21C2, (C2×C4).1576(C2×D4), (C2×C4).688(C4○D4), (C2×C4).467(C22×C4), (C2×C4○D4).289C22, SmallGroup(128,1671)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C4×C4○D8
C1C2C22C2×C4C22×C4C2×C42C4×C4○D4 — C4×C4○D8
C1C2C4 — C4×C4○D8
C1C42C2×C42 — C4×C4○D8
C1C2C2C2×C4 — C4×C4○D8

Generators and relations for C4×C4○D8
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c3 >

Subgroups: 396 in 250 conjugacy classes, 144 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C4 [×10], C22, C22 [×2], C22 [×10], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×26], D4 [×4], D4 [×10], Q8 [×4], Q8 [×2], C23, C23 [×2], C42 [×2], C42 [×2], C42 [×6], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×6], C2×C8 [×2], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C22×C4 [×6], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×8], C4○D4 [×4], C4×C8 [×4], D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C42 [×2], C42⋊C2 [×2], C42⋊C2 [×2], C4×D4 [×4], C4×D4 [×4], C4×Q8 [×4], C22×C8 [×2], C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×8], C2×C4○D4 [×2], C2×C4×C8, C23.24D4 [×2], C23.25D4, C4×D8 [×2], C4×SD16 [×4], C4×Q16 [×2], C4×C4○D4 [×2], C2×C4○D8, C4×C4○D8
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C4○D8 [×4], C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, C2×C4○D8 [×2], C4×C4○D8

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

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A···4L4M···4R4S···4AD8A···8P
order12222222224···44···44···48···8
size11112244441···12···24···42···2

56 irreducible representations

dim11111111112222
type+++++++++++
imageC1C2C2C2C2C2C2C2C2C4D4D4C4○D4C4○D8
kernelC4×C4○D8C2×C4×C8C23.24D4C23.25D4C4×D8C4×SD16C4×Q16C4×C4○D4C2×C4○D8C4○D8C42C22×C4C2×C4C4
# reps1121242211622416

Matrix representation of C4×C4○D8 in GL3(𝔽17) generated by

400
010
001
,
100
0130
0013
,
100
0314
033
,
100
0143
033
G:=sub<GL(3,GF(17))| [4,0,0,0,1,0,0,0,1],[1,0,0,0,13,0,0,0,13],[1,0,0,0,3,3,0,14,3],[1,0,0,0,14,3,0,3,3] >;

C4×C4○D8 in GAP, Magma, Sage, TeX

C_4\times C_4\circ D_8
% in TeX

G:=Group("C4xC4oD8");
// GroupNames label

G:=SmallGroup(128,1671);
// by ID

G=gap.SmallGroup(128,1671);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,248,2804,1411,172]);
// Polycyclic

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

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