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G = C2×C42.6C4order 128 = 27

Direct product of C2 and C42.6C4

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

Aliases: C2×C42.6C4, C42.676C23, C23.35M4(2), C4⋊C881C22, (C23×C4).35C4, (C2×C42).56C4, C8⋊C452C22, C4(C42.6C4), (C2×C8).396C23, C42.301(C2×C4), (C2×C4).637C24, C24.129(C2×C4), (C2×C4).84M4(2), C4.53(C2×M4(2)), (C22×C42).33C2, C2.8(C22×M4(2)), C4.81(C42⋊C2), C22⋊C8.225C22, (C22×C8).429C22, C23.224(C22×C4), C22.165(C23×C4), (C23×C4).658C22, C22.27(C2×M4(2)), (C2×C42).1105C22, (C22×C4).1271C23, C22.70(C42⋊C2), (C2×C4⋊C8)⋊44C2, (C2×C8⋊C4)⋊30C2, C4.288(C2×C4○D4), (C2×C22⋊C8).45C2, (C2×C4).953(C4○D4), (C2×C4)(C42.6C4), (C22×C4).459(C2×C4), (C2×C4).497(C22×C4), C2.37(C2×C42⋊C2), SmallGroup(128,1650)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2×C42.6C4
C1C2C4C2×C4C22×C4C2×C42C22×C42 — C2×C42.6C4
C1C22 — C2×C42.6C4
C1C22×C4 — C2×C42.6C4
C1C2C2C2×C4 — C2×C42.6C4

Generators and relations for C2×C42.6C4
 G = < a,b,c,d | a2=b4=c4=1, d4=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c2, dcd-1=b2c >

Subgroups: 332 in 244 conjugacy classes, 156 normal (24 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×8], C4 [×8], C22, C22 [×10], C22 [×12], C8 [×8], C2×C4 [×2], C2×C4 [×18], C2×C4 [×32], C23, C23 [×6], C23 [×4], C42 [×16], C2×C8 [×8], C2×C8 [×8], C22×C4 [×6], C22×C4 [×12], C22×C4 [×12], C24, C8⋊C4 [×8], C22⋊C8 [×8], C4⋊C8 [×8], C2×C42 [×4], C2×C42 [×8], C22×C8 [×4], C23×C4 [×3], C2×C8⋊C4 [×2], C2×C22⋊C8 [×2], C2×C4⋊C8 [×2], C42.6C4 [×8], C22×C42, C2×C42.6C4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], M4(2) [×8], C22×C4 [×14], C4○D4 [×4], C24, C42⋊C2 [×4], C2×M4(2) [×12], C23×C4, C2×C4○D4 [×2], C42.6C4 [×4], C2×C42⋊C2, C22×M4(2) [×2], C2×C42.6C4

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

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

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

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

56 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4AB8A···8P
order12···222224···44···48···8
size11···122221···12···24···4

56 irreducible representations

dim11111111222
type++++++
imageC1C2C2C2C2C2C4C4M4(2)C4○D4M4(2)
kernelC2×C42.6C4C2×C8⋊C4C2×C22⋊C8C2×C4⋊C8C42.6C4C22×C42C2×C42C23×C4C2×C4C2×C4C23
# reps122281124888

Matrix representation of C2×C42.6C4 in GL5(𝔽17)

160000
016000
001600
00010
00001
,
160000
016000
013100
00040
00004
,
160000
04000
00400
00040
000013
,
160000
041500
001300
000016
000130

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],[16,0,0,0,0,0,16,13,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[16,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,13],[16,0,0,0,0,0,4,0,0,0,0,15,13,0,0,0,0,0,0,13,0,0,0,16,0] >;

C2×C42.6C4 in GAP, Magma, Sage, TeX

C_2\times C_4^2._6C_4
% in TeX

G:=Group("C2xC4^2.6C4");
// GroupNames label

G:=SmallGroup(128,1650);
// by ID

G=gap.SmallGroup(128,1650);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,100,124]);
// Polycyclic

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

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