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G = C2×Q8.D4order 128 = 27

Direct product of C2 and Q8.D4

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

Aliases: C2×Q8.D4, C42.210D4, C42.319C23, Q8.2(C2×D4), C4⋊C864C22, (C2×Q8).170D4, (C22×Q16)⋊8C2, (C4×Q8)⋊78C22, C4.65(C22×D4), C4.68(C4⋊D4), C4⋊C4.375C23, (C2×C8).134C23, (C2×C4).238C24, (C2×Q16)⋊40C22, (C2×D4).47C23, C23.857(C2×D4), (C22×C4).717D4, (C2×Q8).34C23, Q8⋊C477C22, C22.89(C4○D8), (C2×C42).807C22, (C22×C8).339C22, (C22×SD16).14C2, C22.498(C22×D4), D4⋊C4.154C22, C22.170(C4⋊D4), (C22×C4).1528C23, (C2×SD16).132C22, C4.4D4.124C22, (C22×D4).337C22, (C22×Q8).270C22, C22.103(C8.C22), (C2×C4⋊C8)⋊26C2, (C2×C4×Q8)⋊35C2, C2.11(C2×C4○D8), C4.148(C2×C4○D4), (C2×C4).464(C2×D4), C2.56(C2×C4⋊D4), (C2×Q8⋊C4)⋊38C2, C2.14(C2×C8.C22), (C2×D4⋊C4).24C2, (C2×C4).905(C4○D4), (C2×C4⋊C4).919C22, (C2×C4.4D4).37C2, SmallGroup(128,1766)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×Q8.D4
C1C2C4C2×C4C22×C4C22×Q8C2×C4×Q8 — C2×Q8.D4
C1C2C2×C4 — C2×Q8.D4
C1C23C2×C42 — C2×Q8.D4
C1C2C2C2×C4 — C2×Q8.D4

Generators and relations for C2×Q8.D4
 G = < a,b,c,d,e | a2=b4=d4=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, cd=dc, ece-1=b-1c, ede-1=b2d-1 >

Subgroups: 476 in 250 conjugacy classes, 108 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×2], C4 [×2], C4 [×12], C22, C22 [×6], C22 [×10], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×18], D4 [×6], Q8 [×4], Q8 [×12], C23, C23 [×8], C42 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×4], C2×C8 [×4], SD16 [×8], Q16 [×8], C22×C4 [×3], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C2×Q8 [×8], C2×Q8 [×8], C24, D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×4], C2×C42, C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4, C4×Q8 [×4], C4×Q8 [×2], C4.4D4 [×4], C4.4D4 [×2], C22×C8 [×2], C2×SD16 [×4], C2×SD16 [×4], C2×Q16 [×4], C2×Q16 [×4], C22×D4, C22×Q8 [×2], C2×D4⋊C4, C2×Q8⋊C4, C2×C4⋊C8, Q8.D4 [×8], C2×C4×Q8, C2×C4.4D4, C22×SD16, C22×Q16, C2×Q8.D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C4○D8 [×2], C8.C22 [×2], C22×D4 [×2], C2×C4○D4, Q8.D4 [×4], C2×C4⋊D4, C2×C4○D8, C2×C8.C22, C2×Q8.D4

Smallest permutation representation of C2×Q8.D4
On 64 points
Generators in S64
(1 13)(2 14)(3 15)(4 16)(5 11)(6 12)(7 9)(8 10)(17 31)(18 32)(19 29)(20 30)(21 27)(22 28)(23 25)(24 26)(33 47)(34 48)(35 45)(36 46)(37 43)(38 44)(39 41)(40 42)(49 63)(50 64)(51 61)(52 62)(53 59)(54 60)(55 57)(56 58)
(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 27 3 25)(2 26 4 28)(5 29 7 31)(6 32 8 30)(9 17 11 19)(10 20 12 18)(13 21 15 23)(14 24 16 22)(33 57 35 59)(34 60 36 58)(37 61 39 63)(38 64 40 62)(41 49 43 51)(42 52 44 50)(45 53 47 55)(46 56 48 54)
(1 45 5 43)(2 46 6 44)(3 47 7 41)(4 48 8 42)(9 39 15 33)(10 40 16 34)(11 37 13 35)(12 38 14 36)(17 63 23 57)(18 64 24 58)(19 61 21 59)(20 62 22 60)(25 55 31 49)(26 56 32 50)(27 53 29 51)(28 54 30 52)
(1 33 3 35)(2 36 4 34)(5 39 7 37)(6 38 8 40)(9 43 11 41)(10 42 12 44)(13 47 15 45)(14 46 16 48)(17 52 19 50)(18 51 20 49)(21 56 23 54)(22 55 24 53)(25 60 27 58)(26 59 28 57)(29 64 31 62)(30 63 32 61)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I4A···4H4I···4R4S4T8A···8H
order12···2224···44···4448···8
size11···1882···24···4884···4

38 irreducible representations

dim111111111222224
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2D4D4D4C4○D4C4○D8C8.C22
kernelC2×Q8.D4C2×D4⋊C4C2×Q8⋊C4C2×C4⋊C8Q8.D4C2×C4×Q8C2×C4.4D4C22×SD16C22×Q16C42C22×C4C2×Q8C2×C4C22C22
# reps111181111224482

Matrix representation of C2×Q8.D4 in GL5(𝔽17)

160000
01000
00100
00010
00001
,
10000
016000
001600
0001615
00011
,
160000
001600
016000
000010
00050
,
160000
00400
04000
00040
00004
,
160000
001300
04000
000130
00044

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,1,0,0,0,15,1],[16,0,0,0,0,0,0,16,0,0,0,16,0,0,0,0,0,0,0,5,0,0,0,10,0],[16,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,4],[16,0,0,0,0,0,0,4,0,0,0,13,0,0,0,0,0,0,13,4,0,0,0,0,4] >;

C2×Q8.D4 in GAP, Magma, Sage, TeX

C_2\times Q_8.D_4
% in TeX

G:=Group("C2xQ8.D4");
// GroupNames label

G:=SmallGroup(128,1766);
// by ID

G=gap.SmallGroup(128,1766);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,352,4037,1027,124]);
// Polycyclic

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

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