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G = C2×C8.12D4order 128 = 27

Direct product of C2 and C8.12D4

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

Aliases: C2×C8.12D4, C42.359D4, C42.713C23, C8.52(C2×D4), (C4×C8)⋊74C22, (C2×C8).261D4, C4.4(C22×D4), (C22×D8).9C2, C4.14(C41D4), (C2×C4).344C24, (C2×C8).593C23, (C2×Q16)⋊44C22, (C22×Q16)⋊11C2, C23.879(C2×D4), (C22×C4).566D4, (C2×Q8).98C23, (C22×SD16)⋊27C2, (C2×SD16)⋊78C22, (C2×D8).129C22, (C2×D4).110C23, C22.98(C4○D8), C4.4D456C22, C22.50(C41D4), (C22×C8).537C22, C22.604(C22×D4), (C2×C42).1128C22, (C22×C4).1559C23, (C22×D4).373C22, (C22×Q8).306C22, (C2×C4×C8)⋊29C2, C2.30(C2×C4○D8), (C2×C4).854(C2×D4), C2.23(C2×C41D4), (C2×C4.4D4)⋊40C2, SmallGroup(128,1878)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×C8.12D4
C1C2C22C2×C4C22×C4C2×C42C2×C4×C8 — C2×C8.12D4
C1C2C2×C4 — C2×C8.12D4
C1C23C2×C42 — C2×C8.12D4
C1C2C2C2×C4 — C2×C8.12D4

Generators and relations for C2×C8.12D4
 G = < a,b,c,d | a2=b8=c4=1, d2=b4, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=b4c-1 >

Subgroups: 628 in 296 conjugacy classes, 116 normal (16 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×6], C22 [×20], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×12], D4 [×12], Q8 [×12], C23, C23 [×16], C42 [×4], C22⋊C4 [×16], C2×C8 [×12], D8 [×8], SD16 [×16], Q16 [×8], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×4], C2×D4 [×10], C2×Q8 [×4], C2×Q8 [×10], C24 [×2], C4×C8 [×4], C2×C42, C2×C22⋊C4 [×4], C4.4D4 [×8], C4.4D4 [×4], C22×C8 [×2], C2×D8 [×4], C2×D8 [×4], C2×SD16 [×8], C2×SD16 [×8], C2×Q16 [×4], C2×Q16 [×4], C22×D4 [×2], C22×Q8 [×2], C2×C4×C8, C8.12D4 [×8], C2×C4.4D4 [×2], C22×D8, C22×SD16 [×2], C22×Q16, C2×C8.12D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C41D4 [×4], C4○D8 [×4], C22×D4 [×3], C8.12D4 [×4], C2×C41D4, C2×C4○D8 [×2], C2×C8.12D4

Smallest permutation representation of C2×C8.12D4
On 64 points
Generators in S64
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 53)(10 54)(11 55)(12 56)(13 49)(14 50)(15 51)(16 52)(17 45)(18 46)(19 47)(20 48)(21 41)(22 42)(23 43)(24 44)(33 62)(34 63)(35 64)(36 57)(37 58)(38 59)(39 60)(40 61)
(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 19 11 63)(2 20 12 64)(3 21 13 57)(4 22 14 58)(5 23 15 59)(6 24 16 60)(7 17 9 61)(8 18 10 62)(25 47 55 34)(26 48 56 35)(27 41 49 36)(28 42 50 37)(29 43 51 38)(30 44 52 39)(31 45 53 40)(32 46 54 33)
(1 59 5 63)(2 62 6 58)(3 57 7 61)(4 60 8 64)(9 17 13 21)(10 20 14 24)(11 23 15 19)(12 18 16 22)(25 38 29 34)(26 33 30 37)(27 36 31 40)(28 39 32 35)(41 53 45 49)(42 56 46 52)(43 51 47 55)(44 54 48 50)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4L4M4N4O4P8A···8P
order12···222224···444448···8
size11···188882···288882···2

44 irreducible representations

dim11111112222
type++++++++++
imageC1C2C2C2C2C2C2D4D4D4C4○D8
kernelC2×C8.12D4C2×C4×C8C8.12D4C2×C4.4D4C22×D8C22×SD16C22×Q16C42C2×C8C22×C4C22
# reps118212128216

Matrix representation of C2×C8.12D4 in GL5(𝔽17)

160000
016000
001600
00010
00001
,
10000
00100
016000
0001010
000120
,
10000
00100
016000
00040
00004
,
160000
00100
01000
00040
0001313

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,10,12,0,0,0,10,0],[1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,4],[16,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,4,13,0,0,0,0,13] >;

C2×C8.12D4 in GAP, Magma, Sage, TeX

C_2\times C_8._{12}D_4
% in TeX

G:=Group("C2xC8.12D4");
// GroupNames label

G:=SmallGroup(128,1878);
// by ID

G=gap.SmallGroup(128,1878);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,520,2804,172]);
// Polycyclic

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

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