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## G = C2×Q8⋊D4order 128 = 27

### Direct product of C2 and Q8⋊D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×Q8⋊D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — Q8×C23 — C2×Q8⋊D4
 Lower central C1 — C2 — C2×C4 — C2×Q8⋊D4
 Upper central C1 — C23 — C23×C4 — C2×Q8⋊D4
 Jennings C1 — C2 — C2 — C2×C4 — C2×Q8⋊D4

Generators and relations for C2×Q8⋊D4
G = < a,b,c,d,e | a2=b4=d4=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=ebe=b-1, dcd-1=ece=b-1c, ede=d-1 >

Subgroups: 764 in 396 conjugacy classes, 124 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×4], C4 [×12], C22, C22 [×10], C22 [×22], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×42], D4 [×14], Q8 [×8], Q8 [×28], C23, C23 [×6], C23 [×12], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4, C2×C8 [×4], C2×C8 [×4], SD16 [×16], C22×C4 [×2], C22×C4 [×4], C22×C4 [×19], C2×D4 [×2], C2×D4 [×13], C2×Q8 [×12], C2×Q8 [×50], C24, C24, C22⋊C8 [×4], Q8⋊C4 [×8], C2×C22⋊C4, C2×C4⋊C4, C4⋊D4 [×4], C4⋊D4 [×2], C22×C8 [×2], C2×SD16 [×8], C2×SD16 [×8], C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8 [×6], C22×Q8 [×11], C2×C22⋊C8, C2×Q8⋊C4 [×2], Q8⋊D4 [×8], C2×C4⋊D4, C22×SD16 [×2], Q8×C23, C2×Q8⋊D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], SD16 [×4], C2×D4 [×18], C24, C22≀C2 [×4], C2×SD16 [×6], C8.C22 [×2], C22×D4 [×3], Q8⋊D4 [×4], C2×C22≀C2, C22×SD16, C2×C8.C22, C2×Q8⋊D4

Smallest permutation representation of C2×Q8⋊D4
On 64 points
Generators in S64
(1 16)(2 13)(3 14)(4 15)(5 56)(6 53)(7 54)(8 55)(9 17)(10 18)(11 19)(12 20)(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 56 3 54)(2 55 4 53)(5 14 7 16)(6 13 8 15)(9 58 11 60)(10 57 12 59)(17 64 19 62)(18 63 20 61)(21 41 23 43)(22 44 24 42)(25 40 27 38)(26 39 28 37)(29 49 31 51)(30 52 32 50)(33 47 35 45)(34 46 36 48)
(1 25 11 22)(2 28 12 21)(3 27 9 24)(4 26 10 23)(5 51 62 47)(6 50 63 46)(7 49 64 45)(8 52 61 48)(13 31 20 33)(14 30 17 36)(15 29 18 35)(16 32 19 34)(37 60 41 56)(38 59 42 55)(39 58 43 54)(40 57 44 53)
(1 17)(2 20)(3 19)(4 18)(5 59)(6 58)(7 57)(8 60)(9 16)(10 15)(11 14)(12 13)(21 33)(22 36)(23 35)(24 34)(25 30)(26 29)(27 32)(28 31)(37 52)(38 51)(39 50)(40 49)(41 48)(42 47)(43 46)(44 45)(53 64)(54 63)(55 62)(56 61)

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 SD16 C8.C22 kernel C2×Q8⋊D4 C2×C22⋊C8 C2×Q8⋊C4 Q8⋊D4 C2×C4⋊D4 C22×SD16 Q8×C23 C22×C4 C2×Q8 C24 C23 C22 # reps 1 1 2 8 1 2 1 3 8 1 8 2

Matrix representation of C2×Q8⋊D4 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 16 0 0 0 0 0 0 16
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 15 0 0 0 0 1 16
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 10 7 0 0 0 0 5 7
,
 0 16 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 1 0 0 0 0 0 0 0 16 0 0 0 0 0 16 1
,
 1 0 0 0 0 0 0 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 1 16

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,15,16],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,10,5,0,0,0,0,7,7],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,16,16,0,0,0,0,0,1],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,16] >;

C2×Q8⋊D4 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes D_4
% in TeX

G:=Group("C2xQ8:D4");
// GroupNames label

G:=SmallGroup(128,1730);
// by ID

G=gap.SmallGroup(128,1730);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,2804,1411,172]);
// Polycyclic

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

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