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## G = C2×C8⋊3D4order 128 = 27

### Direct product of C2 and C8⋊3D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C8⋊3D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C2×C8⋊C4 — C2×C8⋊3D4
 Lower central C1 — C2 — C2×C4 — C2×C8⋊3D4
 Upper central C1 — C23 — C2×C42 — C2×C8⋊3D4
 Jennings C1 — C2 — C2 — C2×C4 — C2×C8⋊3D4

Generators and relations for C2×C83D4
G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=c-1 >

Subgroups: 804 in 334 conjugacy classes, 116 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C2×C8, D8, SD16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C8⋊C4, C2×C42, C2×C22⋊C4, C4.4D4, C4.4D4, C41D4, C41D4, C22×C8, C2×D8, C2×D8, C2×SD16, C2×SD16, C22×D4, C22×D4, C22×D4, C22×Q8, C2×C8⋊C4, C83D4, C2×C4.4D4, C2×C41D4, C22×D8, C22×SD16, C2×C83D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C41D4, C8⋊C22, C22×D4, C83D4, C2×C41D4, C2×C8⋊C22, C2×C83D4

Smallest permutation representation of C2×C83D4
On 64 points
Generators in S64
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 49)(8 50)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 31)(18 32)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(33 63)(34 64)(35 57)(36 58)(37 59)(38 60)(39 61)(40 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 63 23 9)(2 60 24 14)(3 57 17 11)(4 62 18 16)(5 59 19 13)(6 64 20 10)(7 61 21 15)(8 58 22 12)(25 45 55 37)(26 42 56 34)(27 47 49 39)(28 44 50 36)(29 41 51 33)(30 46 52 38)(31 43 53 35)(32 48 54 40)
(1 55)(2 54)(3 53)(4 52)(5 51)(6 50)(7 49)(8 56)(9 37)(10 36)(11 35)(12 34)(13 33)(14 40)(15 39)(16 38)(17 31)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(24 32)(41 59)(42 58)(43 57)(44 64)(45 63)(46 62)(47 61)(48 60)

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

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

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

32 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2M 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 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 1 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 C8⋊C22 kernel C2×C8⋊3D4 C2×C8⋊C4 C8⋊3D4 C2×C4.4D4 C2×C4⋊1D4 C22×D8 C22×SD16 C42 C2×C8 C22×C4 C22 # reps 1 1 8 1 1 2 2 2 8 2 4

Matrix representation of C2×C83D4 in GL8(𝔽17)

 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 16 1 0 0 0 0 0 0 15 1 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 13 15 0 11 0 0 0 0 1 0 3 0 0 0 0 0 13 6 0 2 0 0 0 0 5 13 16 4
,
 1 16 0 0 0 0 0 0 2 16 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 16 0 16 16 0 0 0 0 16 0 0 0 0 0 0 0 1 16 1 0
,
 1 0 0 0 0 0 0 0 2 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 2 0 1 1

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,15,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,13,1,13,5,0,0,0,0,15,0,6,13,0,0,0,0,0,3,0,16,0,0,0,0,11,0,2,4],[1,2,0,0,0,0,0,0,16,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,16,16,1,0,0,0,0,0,0,0,16,0,0,0,0,16,16,0,1,0,0,0,0,0,16,0,0],[1,2,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,16,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,0,1] >;

C2×C83D4 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes_3D_4
% in TeX

G:=Group("C2xC8:3D4");
// GroupNames label

G:=SmallGroup(128,1880);
// by ID

G=gap.SmallGroup(128,1880);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,723,184,2804,172]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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