Copied to
clipboard

G = C2×C4.C42order 128 = 27

Direct product of C2 and C4.C42

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

Aliases: C2×C4.C42, C24.18Q8, (C23×C8).5C2, (C2×C4).67C42, (C22×C8).28C4, C4.19(C2×C42), (C22×C4).88Q8, C23.75(C2×Q8), C23.62(C4⋊C4), C4(C4.C42), (C22×C4).754D4, (C2×M4(2)).25C4, M4(2).27(C2×C4), (C23×C4).669C22, (C22×C8).467C22, C22.12(C8.C4), C4.19(C2.C42), (C22×C4).1303C23, (C22×M4(2)).13C2, (C2×M4(2)).297C22, C22.30(C2.C42), (C2×C8).201(C2×C4), C2.4(C2×C8.C4), C22.10(C2×C4⋊C4), (C2×C4).124(C4⋊C4), (C2×C4).1294(C2×D4), C4.104(C2×C22⋊C4), (C2×C4)(C4.C42), (C2×C4).347(C22×C4), (C22×C4).403(C2×C4), (C2×C4).394(C22⋊C4), C2.14(C2×C2.C42), SmallGroup(128,469)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C2×C4.C42
C1C2C4C2×C4C22×C4C23×C4C23×C8 — C2×C4.C42
C1C2C4 — C2×C4.C42
C1C22×C4C23×C4 — C2×C4.C42
C1C2C2C22×C4 — C2×C4.C42

Generators and relations for C2×C4.C42
 G = < a,b,c,d | a2=b4=1, c4=d4=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c >

Subgroups: 276 in 196 conjugacy classes, 116 normal (22 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×2], C4 [×6], C22, C22 [×10], C22 [×12], C8 [×12], C2×C4 [×4], C2×C4 [×24], C23 [×3], C23 [×4], C23 [×4], C2×C8 [×4], C2×C8 [×24], M4(2) [×8], M4(2) [×12], C22×C4 [×6], C22×C4 [×8], C24, C22×C8 [×6], C22×C8 [×6], C2×M4(2) [×12], C2×M4(2) [×6], C23×C4, C4.C42 [×4], C23×C8, C22×M4(2) [×2], C2×C4.C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×6], Q8 [×2], C23, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C2.C42 [×8], C8.C4 [×4], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C4.C42 [×4], C2×C2.C42, C2×C8.C4 [×2], C2×C4.C42

Smallest permutation representation of C2×C4.C42
On 64 points
Generators in S64
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 45)(10 46)(11 47)(12 48)(13 41)(14 42)(15 43)(16 44)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 49)(33 64)(34 57)(35 58)(36 59)(37 60)(38 61)(39 62)(40 63)
(1 27 5 31)(2 32 6 28)(3 29 7 25)(4 26 8 30)(9 43 13 47)(10 48 14 44)(11 45 15 41)(12 42 16 46)(17 52 21 56)(18 49 22 53)(19 54 23 50)(20 51 24 55)(33 58 37 62)(34 63 38 59)(35 60 39 64)(36 57 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 59 9 49 5 63 13 53)(2 35 14 19 6 39 10 23)(3 61 11 51 7 57 15 55)(4 37 16 21 8 33 12 17)(18 27 34 43 22 31 38 47)(20 29 36 45 24 25 40 41)(26 62 46 56 30 58 42 52)(28 64 48 50 32 60 44 54)

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

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

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

56 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I4J4K4L8A···8P8Q···8AF
order12···222224···444448···88···8
size11···122221···122222···24···4

56 irreducible representations

dim1111112222
type+++++--
imageC1C2C2C2C4C4D4Q8Q8C8.C4
kernelC2×C4.C42C4.C42C23×C8C22×M4(2)C22×C8C2×M4(2)C22×C4C22×C4C24C22
# reps141281661116

Matrix representation of C2×C4.C42 in GL4(𝔽17) generated by

16000
01600
00160
00016
,
1000
0100
0040
00013
,
1000
0100
00016
00130
,
4000
01600
0020
0008
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,13],[1,0,0,0,0,1,0,0,0,0,0,13,0,0,16,0],[4,0,0,0,0,16,0,0,0,0,2,0,0,0,0,8] >;

C2×C4.C42 in GAP, Magma, Sage, TeX

C_2\times C_4.C_4^2
% in TeX

G:=Group("C2xC4.C4^2");
// GroupNames label

G:=SmallGroup(128,469);
// by ID

G=gap.SmallGroup(128,469);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,2019,248,172,4037,124]);
// Polycyclic

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

׿
×
𝔽