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## G = C2×C8○2M4(2)  order 128 = 27

### Direct product of C2 and C8○2M4(2)

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C2×C8○2M4(2)
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C2×C42⋊C2 — C2×C8○2M4(2)
 Lower central C1 — C2 — C2×C8○2M4(2)
 Upper central C1 — C22×C8 — C2×C8○2M4(2)
 Jennings C1 — C2 — C2 — C2×C4 — C2×C8○2M4(2)

Generators and relations for C2×C82M4(2)
G = < a,b,c,d | a2=b8=d2=1, c4=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >

Subgroups: 332 in 296 conjugacy classes, 260 normal (18 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, C2×C8, M4(2), C22×C4, C22×C4, C24, C4×C8, C8⋊C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C22×C8, C22×C8, C2×M4(2), C23×C4, C2×C4×C8, C2×C8⋊C4, C82M4(2), C2×C42⋊C2, C23×C8, C22×M4(2), C2×C82M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, C24, C2×C42, C8○D4, C23×C4, C82M4(2), C22×C42, C2×C8○D4, C2×C82M4(2)

Smallest permutation representation of C2×C82M4(2)
On 64 points
Generators in S64
(1 55)(2 56)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)(25 60)(26 61)(27 62)(28 63)(29 64)(30 57)(31 58)(32 59)(33 44)(34 45)(35 46)(36 47)(37 48)(38 41)(39 42)(40 43)
(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 58 24 38 5 62 20 34)(2 59 17 39 6 63 21 35)(3 60 18 40 7 64 22 36)(4 61 19 33 8 57 23 37)(9 45 55 31 13 41 51 27)(10 46 56 32 14 42 52 28)(11 47 49 25 15 43 53 29)(12 48 50 26 16 44 54 30)
(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(57 61)(58 62)(59 63)(60 64)

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

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

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

80 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4AB 8A ··· 8P 8Q ··· 8AN order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

80 irreducible representations

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

Matrix representation of C2×C82M4(2) in GL4(𝔽17) generated by

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

C2×C82M4(2) in GAP, Magma, Sage, TeX

C_2\times C_8\circ_2M_4(2)
% in TeX

G:=Group("C2xC8o2M4(2)");
// GroupNames label

G:=SmallGroup(128,1604);
// by ID

G=gap.SmallGroup(128,1604);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,232,723,172]);
// Polycyclic

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

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