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## G = C2×D4.C8order 128 = 27

### Direct product of C2 and D4.C8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×D4.C8
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C22×C8 — C2×C8○D4 — C2×D4.C8
 Lower central C1 — C2 — C4 — C2×D4.C8
 Upper central C1 — C2×C8 — C22×C8 — C2×D4.C8
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C2×D4.C8

Generators and relations for C2×D4.C8
G = < a,b,c,d | a2=b4=c2=1, d8=b2, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=bc >

Subgroups: 172 in 112 conjugacy classes, 60 normal (36 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C16, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C2×C16, C2×C16, M5(2), M5(2), C22×C8, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C8○D4, C2×C4○D4, D4.C8, C22×C16, C2×M5(2), C2×C8○D4, C2×D4.C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C22⋊C8, C2×C22⋊C4, C22×C8, C2×M4(2), D4.C8, C2×C22⋊C8, C2×D4.C8

Smallest permutation representation of C2×D4.C8
On 64 points
Generators in S64
(1 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)(33 62)(34 63)(35 64)(36 49)(37 50)(38 51)(39 52)(40 53)(41 54)(42 55)(43 56)(44 57)(45 58)(46 59)(47 60)(48 61)
(1 56 9 64)(2 57 10 49)(3 58 11 50)(4 59 12 51)(5 60 13 52)(6 61 14 53)(7 62 15 54)(8 63 16 55)(17 36 25 44)(18 37 26 45)(19 38 27 46)(20 39 28 47)(21 40 29 48)(22 41 30 33)(23 42 31 34)(24 43 32 35)
(1 64)(2 10)(3 58)(5 52)(6 14)(7 62)(9 56)(11 50)(13 60)(15 54)(17 25)(18 37)(20 47)(21 29)(22 41)(24 35)(26 45)(28 39)(30 33)(32 43)(34 42)(38 46)(51 59)(55 63)
(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)

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 8A ··· 8H 8I 8J 8K 8L 8M 8N 8O 8P 16A ··· 16P 16Q ··· 16X order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 8 ··· 8 8 8 8 8 8 8 8 8 16 ··· 16 16 ··· 16 size 1 1 1 1 2 2 4 4 1 1 1 1 2 2 4 4 1 ··· 1 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C8 C8 C8 D4 M4(2) M4(2) D4.C8 kernel C2×D4.C8 D4.C8 C22×C16 C2×M5(2) C2×C8○D4 C2×M4(2) C8○D4 C2×C4○D4 C2×D4 C2×Q8 C4○D4 C2×C8 C2×C4 C23 C2 # reps 1 4 1 1 1 2 4 2 4 4 8 4 2 2 16

Matrix representation of C2×D4.C8 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 16 0 0 0 0 16 0 0 0 0 7 4 0 0 13 10
,
 16 1 0 0 0 1 0 0 0 0 7 4 0 0 5 10
,
 13 11 0 0 9 4 0 0 0 0 16 8 0 0 9 5
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,7,13,0,0,4,10],[16,0,0,0,1,1,0,0,0,0,7,5,0,0,4,10],[13,9,0,0,11,4,0,0,0,0,16,9,0,0,8,5] >;

C2×D4.C8 in GAP, Magma, Sage, TeX

C_2\times D_4.C_8
% in TeX

G:=Group("C2xD4.C8");
// GroupNames label

G:=SmallGroup(128,848);
// by ID

G=gap.SmallGroup(128,848);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1018,248,1411,102,124]);
// Polycyclic

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

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