Copied to
clipboard

G = C2×C4×M4(2)  order 128 = 27

Direct product of C2×C4 and M4(2)

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

Aliases: C2×C4×M4(2), C42M4(2), C23.37C42, C42.748C23, C87(C22×C4), C4(C4×M4(2)), (C4×C8)⋊78C22, C4.30(C2×C42), (C2×C4).76C42, (C2×C42).52C4, C4.57(C23×C4), (C23×C4).33C4, C42(C4×M4(2)), C422(C8⋊C4), C8⋊C467C22, C42.331(C2×C4), (C2×C8).610C23, (C2×C4).621C24, C24.123(C2×C4), C422(C2×M4(2)), (C22×C42).29C2, C22.16(C2×C42), C2.13(C22×C42), C22.32(C23×C4), C2.2(C22×M4(2)), C42(C22×M4(2)), (C23×C4).685C22, C23.287(C22×C4), (C22×C8).582C22, C22.58(C2×M4(2)), (C22×C4).1647C23, (C2×C42).1100C22, (C22×M4(2)).32C2, (C2×M4(2)).378C22, (C2×C4×C8)⋊39C2, C42(C2×C8⋊C4), (C2×C8)⋊32(C2×C4), C8⋊C4(C2×C42), (C2×C8⋊C4)⋊40C2, (C2×C4)(C4×M4(2)), (C2×C4)3(C8⋊C4), C422(C2×C8⋊C4), (C2×C42)(C2×M4(2)), (C2×C4).623(C22×C4), (C22×C4).455(C2×C4), (C2×C42)(C22×M4(2)), (C2×C4)2(C2×C8⋊C4), SmallGroup(128,1603)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C2×C4×M4(2)
C1C2C22C2×C4C22×C4C23×C4C22×C42 — C2×C4×M4(2)
C1C2 — C2×C4×M4(2)
C1C2×C42 — C2×C4×M4(2)
C1C2C2C2×C4 — C2×C4×M4(2)

Generators and relations for C2×C4×M4(2)
 G = < a,b,c,d | a2=b4=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c5 >

Subgroups: 380 in 328 conjugacy classes, 276 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×16], C4 [×4], C22, C22 [×10], C22 [×12], C8 [×16], C2×C4 [×2], C2×C4 [×42], C2×C4 [×20], C23, C23 [×6], C23 [×4], C42 [×16], C2×C8 [×24], M4(2) [×32], C22×C4 [×2], C22×C4 [×24], C22×C4 [×8], C24, C4×C8 [×8], C8⋊C4 [×8], C2×C42 [×2], C2×C42 [×10], C22×C8 [×4], C2×M4(2) [×24], C23×C4, C23×C4 [×2], C2×C4×C8 [×2], C2×C8⋊C4 [×2], C4×M4(2) [×8], C22×C42, C22×M4(2) [×2], C2×C4×M4(2)
Quotients: C1, C2 [×15], C4 [×24], C22 [×35], C2×C4 [×84], C23 [×15], C42 [×16], M4(2) [×8], C22×C4 [×42], C24, C2×C42 [×12], C2×M4(2) [×12], C23×C4 [×3], C4×M4(2) [×4], C22×C42, C22×M4(2) [×2], C2×C4×M4(2)

Smallest permutation representation of C2×C4×M4(2)
On 64 points
Generators in S64
(1 61)(2 62)(3 63)(4 64)(5 57)(6 58)(7 59)(8 60)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(17 52)(18 53)(19 54)(20 55)(21 56)(22 49)(23 50)(24 51)(25 48)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)
(1 11 19 32)(2 12 20 25)(3 13 21 26)(4 14 22 27)(5 15 23 28)(6 16 24 29)(7 9 17 30)(8 10 18 31)(33 49 42 64)(34 50 43 57)(35 51 44 58)(36 52 45 59)(37 53 46 60)(38 54 47 61)(39 55 48 62)(40 56 41 63)
(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)(2 24)(3 21)(4 18)(5 23)(6 20)(7 17)(8 22)(9 30)(10 27)(11 32)(12 29)(13 26)(14 31)(15 28)(16 25)(33 46)(34 43)(35 48)(36 45)(37 42)(38 47)(39 44)(40 41)(49 60)(50 57)(51 62)(52 59)(53 64)(54 61)(55 58)(56 63)

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

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

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

80 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4X4Y···4AJ8A···8AF
order12···222224···44···48···8
size11···122221···12···22···2

80 irreducible representations

dim1111111112
type++++++
imageC1C2C2C2C2C2C4C4C4M4(2)
kernelC2×C4×M4(2)C2×C4×C8C2×C8⋊C4C4×M4(2)C22×C42C22×M4(2)C2×C42C2×M4(2)C23×C4C2×C4
# reps1228121232416

Matrix representation of C2×C4×M4(2) in GL4(𝔽17) generated by

16000
01600
00160
00016
,
13000
0100
00160
00016
,
4000
0100
001311
0094
,
1000
01600
00116
00016
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[13,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[4,0,0,0,0,1,0,0,0,0,13,9,0,0,11,4],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,16,16] >;

C2×C4×M4(2) in GAP, Magma, Sage, TeX

C_2\times C_4\times M_4(2)
% in TeX

G:=Group("C2xC4xM4(2)");
// GroupNames label

G:=SmallGroup(128,1603);
// by ID

G=gap.SmallGroup(128,1603);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,232,1430,172]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^5>;
// generators/relations

׿
×
𝔽