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## G = C2×C4⋊M4(2)  order 128 = 27

### Direct product of C2 and C4⋊M4(2)

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C4⋊M4(2)
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×C42 — C2×C4⋊M4(2)
 Lower central C1 — C22 — C2×C4⋊M4(2)
 Upper central C1 — C22×C4 — 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, cbc-1=b-1, bd=db, dcd=c5 >

Subgroups: 380 in 288 conjugacy classes, 196 normal (16 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×2], C4 [×14], C4 [×4], C22, C22 [×10], C22 [×12], C8 [×8], C2×C4 [×2], C2×C4 [×42], C2×C4 [×20], C23, C23 [×6], C23 [×4], C42 [×16], C2×C8 [×8], C2×C8 [×8], M4(2) [×16], C22×C4 [×2], C22×C4 [×24], C22×C4 [×8], C24, C4⋊C8 [×16], C2×C42 [×2], C2×C42 [×10], C22×C8 [×4], C2×M4(2) [×8], C2×M4(2) [×8], C23×C4, C23×C4 [×2], C2×C4⋊C8 [×4], C4⋊M4(2) [×8], C22×C42, C22×M4(2) [×2], C2×C4⋊M4(2)
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C4⋊C4 [×16], M4(2) [×8], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, C2×C4⋊C4 [×12], C2×M4(2) [×12], C23×C4, C22×D4, C22×Q8, C4⋊M4(2) [×4], C22×C4⋊C4, C22×M4(2) [×2], C2×C4⋊M4(2)

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

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4AB 8A ··· 8P 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 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 type + + + + + + - image C1 C2 C2 C2 C2 C4 C4 D4 Q8 M4(2) kernel C2×C4⋊M4(2) C2×C4⋊C8 C4⋊M4(2) C22×C42 C22×M4(2) C2×C42 C23×C4 C22×C4 C22×C4 C2×C4 # reps 1 4 8 1 2 12 4 4 4 16

Matrix representation of C2×C4⋊M4(2) in GL5(𝔽17)

 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16
,
 1 0 0 0 0 0 1 15 0 0 0 1 16 0 0 0 0 0 4 0 0 0 0 0 13
,
 1 0 0 0 0 0 16 2 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 4 0
,
 16 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 16

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

C2×C4⋊M4(2) in GAP, Magma, Sage, TeX

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

G:=Group("C2xC4:M4(2)");
// GroupNames label

G:=SmallGroup(128,1635);
// by ID

G=gap.SmallGroup(128,1635);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,1430,124]);
// 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,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^5>;
// generators/relations

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