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## G = C2×C4×M4(2)  order 128 = 27

### Direct product of C2×C4 and M4(2)

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C2×C4×M4(2)
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C22×C42 — C2×C4×M4(2)
 Lower central C1 — C2 — C2×C4×M4(2)
 Upper central C1 — C2×C42 — C2×C4×M4(2)
 Jennings C1 — C2 — C2 — C2×C4 — C2×C4×M4(2)

Generators and relations for C2×C4×M4(2)
G = < a,b,c,d | a2=b4=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c5 >

Subgroups: 380 in 328 conjugacy classes, 276 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C24, C4×C8, C8⋊C4, C2×C42, C2×C42, C22×C8, C2×M4(2), C23×C4, C23×C4, C2×C4×C8, C2×C8⋊C4, C4×M4(2), C22×C42, C22×M4(2), C2×C4×M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, M4(2), C22×C4, C24, C2×C42, C2×M4(2), C23×C4, C4×M4(2), C22×C42, C22×M4(2), C2×C4×M4(2)

Smallest permutation representation of C2×C4×M4(2)
On 64 points
Generators in S64
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(25 48)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(55 64)(56 57)
(1 11 63 32)(2 12 64 25)(3 13 57 26)(4 14 58 27)(5 15 59 28)(6 16 60 29)(7 9 61 30)(8 10 62 31)(17 37 53 46)(18 38 54 47)(19 39 55 48)(20 40 56 41)(21 33 49 42)(22 34 50 43)(23 35 51 44)(24 36 52 45)
(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)(2 60)(3 57)(4 62)(5 59)(6 64)(7 61)(8 58)(9 30)(10 27)(11 32)(12 29)(13 26)(14 31)(15 28)(16 25)(17 49)(18 54)(19 51)(20 56)(21 53)(22 50)(23 55)(24 52)(33 46)(34 43)(35 48)(36 45)(37 42)(38 47)(39 44)(40 41)

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

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

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

80 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4X 4Y ··· 4AJ 8A ··· 8AF order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 2 ··· 2

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 type + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 M4(2) kernel C2×C4×M4(2) C2×C4×C8 C2×C8⋊C4 C4×M4(2) C22×C42 C22×M4(2) C2×C42 C2×M4(2) C23×C4 C2×C4 # reps 1 2 2 8 1 2 12 32 4 16

Matrix representation of C2×C4×M4(2) in GL4(𝔽17) generated by

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

C2×C4×M4(2) in GAP, Magma, Sage, TeX

C_2\times C_4\times M_4(2)
% in TeX

G:=Group("C2xC4xM4(2)");
// GroupNames label

G:=SmallGroup(128,1603);
// by ID

G=gap.SmallGroup(128,1603);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,232,1430,172]);
// Polycyclic

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

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