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## G = (C2×D4)⋊Q8order 128 = 27

### 1st semidirect product of C2×D4 and Q8 acting via Q8/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — (C2×D4)⋊Q8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×D4 — C24.3C22 — (C2×D4)⋊Q8
 Lower central C1 — C2 — C22×C4 — (C2×D4)⋊Q8
 Upper central C1 — C23 — C2×C42 — (C2×D4)⋊Q8
 Jennings C1 — C2 — C2 — C22×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, dcd-1=ab2c, ece-1=abc, ede-1=d-1 >

Subgroups: 400 in 163 conjugacy classes, 54 normal (28 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×9], C22 [×7], C22 [×10], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×15], D4 [×6], Q8 [×8], C23, C23 [×8], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×6], C22×C4 [×3], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C2×Q8 [×8], C24, D4⋊C4 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4, C4⋊Q8 [×4], C22×C8 [×2], C22×D4, C22×Q8, C22.7C42, C22.4Q16 [×2], C24.3C22, C2×D4⋊C4 [×2], C2×C4⋊Q8, (C2×D4)⋊Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, D8 [×2], SD16 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×3], C22≀C2, C22⋊Q8 [×3], C4.4D4 [×3], C2×D8, C2×SD16, C8⋊C22, C8.C22, C23⋊Q8, C22⋊D8, Q8⋊D4, D4⋊Q8, D42Q8, C4.4D8, C42.28C22, (C2×D4)⋊Q8

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

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

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

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

32 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4N 8A ··· 8H order 1 2 ··· 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 8 8 2 2 2 2 4 4 4 4 8 ··· 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 D4 D4 Q8 D8 SD16 C4○D4 C8⋊C22 C8.C22 kernel (C2×D4)⋊Q8 C22.7C42 C22.4Q16 C24.3C22 C2×D4⋊C4 C2×C4⋊Q8 C4⋊C4 C22×C4 C2×D4 C2×C4 C2×C4 C2×C4 C22 C22 # reps 1 1 2 1 2 1 4 2 2 4 4 6 1 1

Matrix representation of (C2×D4)⋊Q8 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 2 0 0 0 0 16 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 16 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16
,
 1 2 0 0 0 0 16 16 0 0 0 0 0 0 16 2 0 0 0 0 16 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 13 0 0 0 0 0 4 4 0 0 0 0 0 0 0 7 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,1,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[1,16,0,0,0,0,2,16,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[13,4,0,0,0,0,0,4,0,0,0,0,0,0,0,12,0,0,0,0,7,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

(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(128,755);
// by ID

G=gap.SmallGroup(128,755);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,64,422,387,2804,718,172]);
// 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=a*b^2*c,e*c*e^-1=a*b*c,e*d*e^-1=d^-1>;
// generators/relations

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