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G = C2×D4.5D4order 128 = 27

Direct product of C2 and D4.5D4

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

Aliases: C2×D4.5D4, M4(2).9C23, C4○D4.33D4, D4.13(C2×D4), (C2×C8).352D4, C8.113(C2×D4), Q8.13(C2×D4), (C2×D4).225D4, (C2×C4).17C24, (C2×Q8).180D4, (C2×C8).262C23, (C2×Q16)⋊53C22, (C22×Q16)⋊17C2, C8○D4.10C22, C4○D4.29C23, C4.164(C22×D4), (C2×Q8).59C23, C4.115(C4⋊D4), C8.C417C22, C8.C22.2C22, C23.318(C4○D4), C4.10D413C22, C22.91(C4⋊D4), (C22×C4).992C23, (C22×C8).266C22, (C2×M4(2)).60C22, (C22×Q8).286C22, (C2×C8○D4).10C2, C2.88(C2×C4⋊D4), (C2×C8.C4)⋊27C2, (C2×C4).1429(C2×D4), C22.20(C2×C4○D4), (C2×C4).291(C4○D4), (C2×C4.10D4)⋊11C2, (C2×C8.C22).12C2, (C2×C4○D4).306C22, SmallGroup(128,1798)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×D4.5D4
C1C2C4C2×C4C22×C4C2×C4○D4C2×C8○D4 — C2×D4.5D4
C1C2C2×C4 — C2×D4.5D4
C1C22C22×C4 — C2×D4.5D4
C1C2C2C2×C4 — C2×D4.5D4

Generators and relations for C2×D4.5D4
 G = < a,b,c,d,e | a2=b4=c2=1, d4=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=b2d3 >

Subgroups: 380 in 224 conjugacy classes, 100 normal (28 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×6], C22 [×3], C22 [×6], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×13], D4 [×2], D4 [×5], Q8 [×2], Q8 [×13], C23, C23, C2×C8 [×2], C2×C8 [×4], C2×C8 [×7], M4(2) [×6], M4(2) [×7], SD16 [×8], Q16 [×16], C22×C4, C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C2×Q8 [×4], C2×Q8 [×10], C4○D4 [×4], C4○D4 [×2], C4.10D4 [×8], C8.C4 [×4], C22×C8, C22×C8, C2×M4(2), C2×M4(2) [×2], C2×M4(2), C8○D4 [×4], C8○D4 [×2], C2×SD16 [×2], C2×Q16 [×4], C2×Q16 [×6], C8.C22 [×8], C8.C22 [×4], C22×Q8 [×2], C2×C4○D4, C2×C4.10D4 [×2], C2×C8.C4, D4.5D4 [×8], C2×C8○D4, C22×Q16, C2×C8.C22 [×2], C2×D4.5D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, D4.5D4 [×2], C2×C4⋊D4, C2×D4.5D4

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

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E···8J8K8L8M8N
order12222222444444444488888···88888
size11112244222244888822224···48888

32 irreducible representations

dim11111112222224
type+++++++++++-
imageC1C2C2C2C2C2C2D4D4D4D4C4○D4C4○D4D4.5D4
kernelC2×D4.5D4C2×C4.10D4C2×C8.C4D4.5D4C2×C8○D4C22×Q16C2×C8.C22C2×C8C2×D4C2×Q8C4○D4C2×C4C23C2
# reps12181124112224

Matrix representation of C2×D4.5D4 in GL6(𝔽17)

1600000
0160000
001000
000100
000010
000001
,
100000
010000
0011500
0011600
0000115
0000116
,
1600000
0160000
0011500
0001600
0000115
0000016
,
010000
1600000
0014030
0001403
00140140
00014014
,
0160000
1600000
000404
002020
0004013
0020150

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,0,0,0,0,1,1,0,0,0,0,15,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,15,16,0,0,0,0,0,0,1,0,0,0,0,0,15,16],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,14,0,14,0,0,0,0,14,0,14,0,0,3,0,14,0,0,0,0,3,0,14],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,2,0,2,0,0,4,0,4,0,0,0,0,2,0,15,0,0,4,0,13,0] >;

C2×D4.5D4 in GAP, Magma, Sage, TeX

C_2\times D_4._5D_4
% in TeX

G:=Group("C2xD4.5D4");
// GroupNames label

G:=SmallGroup(128,1798);
// by ID

G=gap.SmallGroup(128,1798);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,568,758,2804,172,4037,1027,124]);
// Polycyclic

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

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