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## G = C2×D4.7D4order 128 = 27

### Direct product of C2 and D4.7D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×D4.7D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C22×C4○D4 — C2×D4.7D4
 Lower central C1 — C2 — C2×C4 — C2×D4.7D4
 Upper central C1 — C23 — C23×C4 — C2×D4.7D4
 Jennings C1 — C2 — C2 — C2×C4 — C2×D4.7D4

Generators and relations for C2×D4.7D4
G = < a,b,c,d,e | a2=b4=c2=d4=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=ece-1=bc, ede-1=d-1 >

Subgroups: 716 in 380 conjugacy classes, 116 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×6], C22 [×26], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×40], D4 [×4], D4 [×22], Q8 [×4], Q8 [×12], C23, C23 [×2], C23 [×16], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×4], C2×C8 [×4], SD16 [×8], Q16 [×8], C22×C4 [×2], C22×C4 [×4], C22×C4 [×19], C2×D4 [×6], C2×D4 [×15], C2×Q8 [×8], C2×Q8 [×8], C4○D4 [×32], C24, C24, C22⋊C8 [×4], D4⋊C4 [×4], Q8⋊C4 [×4], C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8 [×4], C22⋊Q8 [×2], C22×C8 [×2], C2×SD16 [×4], C2×SD16 [×4], C2×Q16 [×4], C2×Q16 [×4], C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8 [×2], C2×C4○D4 [×4], C2×C4○D4 [×10], C2×C22⋊C8, C2×D4⋊C4, C2×Q8⋊C4, D4.7D4 [×8], C2×C22⋊Q8, C22×SD16, C22×Q16, C22×C4○D4, C2×D4.7D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C4○D8 [×2], C8.C22 [×2], C22×D4 [×3], D4.7D4 [×4], C2×C22≀C2, C2×C4○D8, C2×C8.C22, C2×D4.7D4

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

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2M 4A ··· 4H 4I 4J 4K 4L 4M 4N 4O 4P 8A ··· 8H order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 4 4 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 4 ··· 4 2 ··· 2 4 4 4 4 8 8 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 C4○D8 C8.C22 kernel C2×D4.7D4 C2×C22⋊C8 C2×D4⋊C4 C2×Q8⋊C4 D4.7D4 C2×C22⋊Q8 C22×SD16 C22×Q16 C22×C4○D4 C22×C4 C2×D4 C2×Q8 C24 C22 C22 # reps 1 1 1 1 8 1 1 1 1 3 4 4 1 8 2

Matrix representation of C2×D4.7D4 in GL5(𝔽17)

 16 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16
,
 1 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 16 0
,
 16 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0
,
 16 0 0 0 0 0 0 2 0 0 0 8 0 0 0 0 0 0 14 3 0 0 0 3 3
,
 1 0 0 0 0 0 0 15 0 0 0 8 0 0 0 0 0 0 12 5 0 0 0 5 5

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,1,0],[16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0],[16,0,0,0,0,0,0,8,0,0,0,2,0,0,0,0,0,0,14,3,0,0,0,3,3],[1,0,0,0,0,0,0,8,0,0,0,15,0,0,0,0,0,0,12,5,0,0,0,5,5] >;

C2×D4.7D4 in GAP, Magma, Sage, TeX

C_2\times D_4._7D_4
% in TeX

G:=Group("C2xD4.7D4");
// GroupNames label

G:=SmallGroup(128,1733);
// by ID

G=gap.SmallGroup(128,1733);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,352,2804,1411,172]);
// Polycyclic

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

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