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## G = C2×D4⋊2Q8order 128 = 27

### Direct product of C2 and D4⋊2Q8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×D4⋊2Q8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C2×C4×D4 — C2×D4⋊2Q8
 Lower central C1 — C2 — C2×C4 — C2×D4⋊2Q8
 Upper central C1 — C23 — C2×C42 — C2×D4⋊2Q8
 Jennings C1 — C2 — C2 — C2×C4 — C2×D4⋊2Q8

Generators and relations for C2×D42Q8
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, dcd-1=b2c, 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], C4.Q8 [×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×C4.Q8 [×2], D42Q8 [×8], C2×C4×D4, C2×C4⋊Q8, C2×D42Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], SD16 [×4], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C22⋊Q8 [×4], C2×SD16 [×6], C8⋊C22 [×2], C22×D4, C22×Q8, C2×C4○D4, D42Q8 [×4], C2×C22⋊Q8, C22×SD16, C2×C8⋊C22, C2×D42Q8

Smallest permutation representation of C2×D42Q8
On 64 points
Generators in S64
(1 15)(2 16)(3 13)(4 14)(5 9)(6 10)(7 11)(8 12)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)(33 45)(34 46)(35 47)(36 48)(37 41)(38 42)(39 43)(40 44)(49 61)(50 62)(51 63)(52 64)(53 57)(54 58)(55 59)(56 60)
(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 32 5 26)(2 29 6 27)(3 30 7 28)(4 31 8 25)(9 22 15 20)(10 23 16 17)(11 24 13 18)(12 21 14 19)(33 58 39 64)(34 59 40 61)(35 60 37 62)(36 57 38 63)(41 50 47 56)(42 51 48 53)(43 52 45 54)(44 49 46 55)
(1 47 5 41)(2 46 6 44)(3 45 7 43)(4 48 8 42)(9 37 15 35)(10 40 16 34)(11 39 13 33)(12 38 14 36)(17 61 23 59)(18 64 24 58)(19 63 21 57)(20 62 22 60)(25 53 31 51)(26 56 32 50)(27 55 29 49)(28 54 30 52)

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

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

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

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 SD16 C4○D4 C8⋊C22 kernel C2×D4⋊2Q8 C2×D4⋊C4 C2×C4⋊C8 C2×C4.Q8 D4⋊2Q8 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×D42Q8 in GL5(𝔽17)

 16 0 0 0 0 0 16 0 0 0 0 0 16 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 16 1 0 0 0 0 0 0 1 0 0 0 1 0
,
 1 0 0 0 0 0 4 0 0 0 0 4 13 0 0 0 0 0 0 16 0 0 0 1 0
,
 1 0 0 0 0 0 1 15 0 0 0 1 16 0 0 0 0 0 12 5 0 0 0 5 5

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[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],[16,0,0,0,0,0,16,16,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,4,4,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,16,0],[1,0,0,0,0,0,1,1,0,0,0,15,16,0,0,0,0,0,12,5,0,0,0,5,5] >;

C2×D42Q8 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes_2Q_8
% in TeX

G:=Group("C2xD4:2Q8");
// GroupNames label

G:=SmallGroup(128,1803);
// by ID

G=gap.SmallGroup(128,1803);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,120,758,4037,1027,124]);
// 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,d*c*d^-1=b^2*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations

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