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## G = Q8×C4○D4order 128 = 27

### Direct product of Q8 and C4○D4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for Q8×C4○D4
G = < a,b,c,d,e | a4=c4=e2=1, b2=a2, d2=c2, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d >

Subgroups: 660 in 534 conjugacy classes, 438 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C22×Q8, C2×C4○D4, C2×C4×Q8, C4×C4○D4, C23.37C23, D4×Q8, D43Q8, Q83Q8, Q82, Q8×C4○D4
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, C24, C22×Q8, C2×C4○D4, C25, Q8×C23, C22×C4○D4, C2.C25, Q8×C4○D4

Smallest permutation representation of Q8×C4○D4
On 64 points
Generators in S64
(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 44 3 42)(2 43 4 41)(5 38 7 40)(6 37 8 39)(9 52 11 50)(10 51 12 49)(13 19 15 17)(14 18 16 20)(21 55 23 53)(22 54 24 56)(25 61 27 63)(26 64 28 62)(29 57 31 59)(30 60 32 58)(33 47 35 45)(34 46 36 48)
(1 13 27 9)(2 14 28 10)(3 15 25 11)(4 16 26 12)(5 34 29 23)(6 35 30 24)(7 36 31 21)(8 33 32 22)(17 61 50 42)(18 62 51 43)(19 63 52 44)(20 64 49 41)(37 45 60 56)(38 46 57 53)(39 47 58 54)(40 48 59 55)
(1 11 27 15)(2 12 28 16)(3 9 25 13)(4 10 26 14)(5 36 29 21)(6 33 30 22)(7 34 31 23)(8 35 32 24)(17 44 50 63)(18 41 51 64)(19 42 52 61)(20 43 49 62)(37 47 60 54)(38 48 57 55)(39 45 58 56)(40 46 59 53)
(1 21)(2 22)(3 23)(4 24)(5 15)(6 16)(7 13)(8 14)(9 31)(10 32)(11 29)(12 30)(17 38)(18 39)(19 40)(20 37)(25 34)(26 35)(27 36)(28 33)(41 56)(42 53)(43 54)(44 55)(45 64)(46 61)(47 62)(48 63)(49 60)(50 57)(51 58)(52 59)

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4V 4W ··· 4AN order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 4 ··· 4 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 4 type + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 Q8 C4○D4 C2.C25 kernel Q8×C4○D4 C2×C4×Q8 C4×C4○D4 C23.37C23 D4×Q8 D4⋊3Q8 Q8⋊3Q8 Q82 C4○D4 Q8 C2 # reps 1 3 3 9 3 9 3 1 8 8 2

Matrix representation of Q8×C4○D4 in GL4(𝔽5) generated by

 4 0 0 0 0 4 0 0 0 0 4 3 0 0 1 1
,
 1 0 0 0 0 1 0 0 0 0 3 1 0 0 0 2
,
 3 0 0 0 0 3 0 0 0 0 1 0 0 0 0 1
,
 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0 4
,
 0 3 0 0 2 0 0 0 0 0 4 0 0 0 0 4
G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,0,0,0,4,1,0,0,3,1],[1,0,0,0,0,1,0,0,0,0,3,0,0,0,1,2],[3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[2,0,0,0,0,3,0,0,0,0,4,0,0,0,0,4],[0,2,0,0,3,0,0,0,0,0,4,0,0,0,0,4] >;

Q8×C4○D4 in GAP, Magma, Sage, TeX

Q_8\times C_4\circ D_4
% in TeX

G:=Group("Q8xC4oD4");
// GroupNames label

G:=SmallGroup(128,2210);
// by ID

G=gap.SmallGroup(128,2210);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,1430,352,570,102]);
// Polycyclic

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

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