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## G = C2×D4⋊Q8order 128 = 27

### Direct product of C2 and D4⋊Q8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×D4⋊Q8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C2×C4×D4 — C2×D4⋊Q8
 Lower central C1 — C2 — C2×C4 — C2×D4⋊Q8
 Upper central C1 — C23 — C2×C42 — C2×D4⋊Q8
 Jennings C1 — C2 — C2 — C2×C4 — C2×D4⋊Q8

Generators and relations for C2×D4⋊Q8
G = < a,b,c,d,e | a2=b4=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 476 in 240 conjugacy classes, 116 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×6], C4 [×8], C22, C22 [×6], C22 [×16], C8 [×4], C2×C4 [×2], C2×C4 [×12], C2×C4 [×22], D4 [×4], D4 [×6], Q8 [×8], C23, C23 [×10], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×6], C4⋊C4 [×7], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×11], C2×D4 [×6], C2×D4 [×3], C2×Q8 [×8], C24, D4⋊C4 [×8], C4⋊C8 [×4], C2.D8 [×8], C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4, C4×D4 [×4], C4×D4 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C22×C8 [×2], C23×C4, C22×D4, C22×Q8, C2×D4⋊C4 [×2], C2×C4⋊C8, C2×C2.D8 [×2], D4⋊Q8 [×8], C2×C4×D4, C2×C4⋊Q8, C2×D4⋊Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C22⋊Q8 [×4], C2×D8 [×6], C8.C22 [×2], C22×D4, C22×Q8, C2×C4○D4, D4⋊Q8 [×4], C2×C22⋊Q8, C22×D8, C2×C8.C22, C2×D4⋊Q8

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 D4 D4 Q8 D8 C4○D4 C8.C22 kernel C2×D4⋊Q8 C2×D4⋊C4 C2×C4⋊C8 C2×C2.D8 D4⋊Q8 C2×C4×D4 C2×C4⋊Q8 C42 C22×C4 C2×D4 C2×C4 C2×C4 C22 # reps 1 2 1 2 8 1 1 2 2 4 8 4 2

Matrix representation of C2×D4⋊Q8 in GL6(𝔽17)

 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 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 0 1 0 0 0 0 16 0
,
 16 0 0 0 0 0 6 1 0 0 0 0 0 0 16 0 0 0 0 0 13 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 13 0 0 0 0 0 1 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 6 2 0 0 0 0 8 11 0 0 0 0 0 0 4 15 0 0 0 0 0 13 0 0 0 0 0 0 14 3 0 0 0 0 3 3

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[16,6,0,0,0,0,0,1,0,0,0,0,0,0,16,13,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,1,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,8,0,0,0,0,2,11,0,0,0,0,0,0,4,0,0,0,0,0,15,13,0,0,0,0,0,0,14,3,0,0,0,0,3,3] >;

C2×D4⋊Q8 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes Q_8
% in TeX

G:=Group("C2xD4:Q8");
// GroupNames label

G:=SmallGroup(128,1802);
// by ID

G=gap.SmallGroup(128,1802);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,4037,1027,124]);
// Polycyclic

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

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