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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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×D4.5D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2×C8○D4 — C2×D4.5D4
 Lower central C1 — C2 — C2×C4 — C2×D4.5D4
 Upper central C1 — C22 — C22×C4 — C2×D4.5D4
 Jennings C1 — C2 — C2 — C2×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, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C2×C8, C2×C8, C2×C8, M4(2), M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C4.10D4, C8.C4, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C2×M4(2), C8○D4, C8○D4, C2×SD16, C2×Q16, C2×Q16, C8.C22, C8.C22, C22×Q8, C2×C4○D4, C2×C4.10D4, C2×C8.C4, D4.5D4, C2×C8○D4, C22×Q16, C2×C8.C22, C2×D4.5D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, D4.5D4, C2×C4⋊D4, C2×D4.5D4

Smallest permutation representation of C2×D4.5D4
On 64 points
Generators in S64
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(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 55)(34 56)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)
(1 43 5 47)(2 44 6 48)(3 45 7 41)(4 46 8 42)(9 24 13 20)(10 17 14 21)(11 18 15 22)(12 19 16 23)(25 49 29 53)(26 50 30 54)(27 51 31 55)(28 52 32 56)(33 58 37 62)(34 59 38 63)(35 60 39 64)(36 61 40 57)
(1 47)(2 48)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)(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 39 5 35)(2 38 6 34)(3 37 7 33)(4 36 8 40)(9 56 13 52)(10 55 14 51)(11 54 15 50)(12 53 16 49)(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)| (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(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)(33,55)(34,56)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54), (1,43,5,47)(2,44,6,48)(3,45,7,41)(4,46,8,42)(9,24,13,20)(10,17,14,21)(11,18,15,22)(12,19,16,23)(25,49,29,53)(26,50,30,54)(27,51,31,55)(28,52,32,56)(33,58,37,62)(34,59,38,63)(35,60,39,64)(36,61,40,57), (1,47)(2,48)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19)(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), (1,39,5,35)(2,38,6,34)(3,37,7,33)(4,36,8,40)(9,56,13,52)(10,55,14,51)(11,54,15,50)(12,53,16,49)(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( (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(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)(33,55)(34,56)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54), (1,43,5,47)(2,44,6,48)(3,45,7,41)(4,46,8,42)(9,24,13,20)(10,17,14,21)(11,18,15,22)(12,19,16,23)(25,49,29,53)(26,50,30,54)(27,51,31,55)(28,52,32,56)(33,58,37,62)(34,59,38,63)(35,60,39,64)(36,61,40,57), (1,47)(2,48)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19)(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), (1,39,5,35)(2,38,6,34)(3,37,7,33)(4,36,8,40)(9,56,13,52)(10,55,14,51)(11,54,15,50)(12,53,16,49)(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([[(1,12),(2,13),(3,14),(4,15),(5,16),(6,9),(7,10),(8,11),(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),(33,55),(34,56),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54)], [(1,43,5,47),(2,44,6,48),(3,45,7,41),(4,46,8,42),(9,24,13,20),(10,17,14,21),(11,18,15,22),(12,19,16,23),(25,49,29,53),(26,50,30,54),(27,51,31,55),(28,52,32,56),(33,58,37,62),(34,59,38,63),(35,60,39,64),(36,61,40,57)], [(1,47),(2,48),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,20),(10,21),(11,22),(12,23),(13,24),(14,17),(15,18),(16,19),(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)], [(1,39,5,35),(2,38,6,34),(3,37,7,33),(4,36,8,40),(9,56,13,52),(10,55,14,51),(11,54,15,50),(12,53,16,49),(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 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A 8B 8C 8D 8E ··· 8J 8K 8L 8M 8N order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 8 8 8 8 8 ··· 8 8 8 8 8 size 1 1 1 1 2 2 4 4 2 2 2 2 4 4 8 8 8 8 2 2 2 2 4 ··· 4 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 C4○D4 C4○D4 D4.5D4 kernel C2×D4.5D4 C2×C4.10D4 C2×C8.C4 D4.5D4 C2×C8○D4 C22×Q16 C2×C8.C22 C2×C8 C2×D4 C2×Q8 C4○D4 C2×C4 C23 C2 # reps 1 2 1 8 1 1 2 4 1 1 2 2 2 4

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

 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 15 0 0 0 0 1 16 0 0 0 0 0 0 1 15 0 0 0 0 1 16
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 15 0 0 0 0 0 16 0 0 0 0 0 0 1 15 0 0 0 0 0 16
,
 0 1 0 0 0 0 16 0 0 0 0 0 0 0 14 0 3 0 0 0 0 14 0 3 0 0 14 0 14 0 0 0 0 14 0 14
,
 0 16 0 0 0 0 16 0 0 0 0 0 0 0 0 4 0 4 0 0 2 0 2 0 0 0 0 4 0 13 0 0 2 0 15 0

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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