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## G = C2×M4(2)⋊C4order 128 = 27

### Direct product of C2 and M4(2)⋊C4

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2×M4(2)⋊C4
G = < a,b,c,d | a2=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b5, dbd-1=b-1, cd=dc >

Subgroups: 412 in 268 conjugacy classes, 180 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C24, C4.Q8, C2.D8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C22×C8, C2×M4(2), C23×C4, C23×C4, C2×C4.Q8, C2×C2.D8, M4(2)⋊C4, C22×C4⋊C4, C2×C42⋊C2, C22×M4(2), C2×M4(2)⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C24, C2×C4⋊C4, C8⋊C22, C8.C22, C23×C4, C22×D4, C22×Q8, M4(2)⋊C4, C22×C4⋊C4, C2×C8⋊C22, C2×C8.C22, C2×M4(2)⋊C4

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 C4 D4 Q8 D4 C8⋊C22 C8.C22 kernel C2×M4(2)⋊C4 C2×C4.Q8 C2×C2.D8 M4(2)⋊C4 C22×C4⋊C4 C2×C42⋊C2 C22×M4(2) C2×M4(2) C22×C4 C22×C4 C24 C22 C22 # reps 1 2 2 8 1 1 1 16 3 4 1 2 2

Matrix representation of C2×M4(2)⋊C4 in GL8(𝔽17)

 16 0 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 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
,
 16 2 0 0 0 0 0 0 16 1 0 0 0 0 0 0 0 0 16 2 0 0 0 0 0 0 16 1 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 0 16 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 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 16 0 0 0 0 0 0 0 0 16
,
 1 15 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 4 9 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 11 4 0 0 0 0 0 0 4 6 0 0 0 0 0 0 0 0 13 11 0 0 0 0 0 0 11 4

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,16,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,16,16,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,9,13,0,0,0,0,0,0,0,0,11,4,0,0,0,0,0,0,4,6,0,0,0,0,0,0,0,0,13,11,0,0,0,0,0,0,11,4] >;

C2×M4(2)⋊C4 in GAP, Magma, Sage, TeX

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

G:=Group("C2xM4(2):C4");
// GroupNames label

G:=SmallGroup(128,1642);
// by ID

G=gap.SmallGroup(128,1642);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,723,2804,172]);
// Polycyclic

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

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