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G = C2×C82M4(2)  order 128 = 27

Direct product of C2 and C82M4(2)

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

Aliases: C2×C82M4(2), C23.38C42, C42.587C23, (C22×C8)⋊21C4, (C4×C8)⋊79C22, C83(C2×M4(2)), (C2×C8)3M4(2), (C2×C4).77C42, C8.46(C22×C4), C4.58(C23×C4), C24.97(C2×C4), C4.31(C2×C42), (C23×C8).25C2, M4(2)(C22×C8), C8(C22×M4(2)), C82(C42⋊C2), C8⋊C468C22, C8(C82M4(2)), C4(C82M4(2)), M4(2)⋊22(C2×C4), (C2×M4(2))⋊21C4, (C2×C8).637C23, (C2×C4).622C24, C42.246(C2×C4), C42⋊C2.39C4, C22.36(C8○D4), C2.14(C22×C42), C22.33(C23×C4), C22.17(C2×C42), (C23×C4).686C22, C23.288(C22×C4), (C22×C8).583C22, (C2×C42).1022C22, (C22×C4).1648C23, (C22×M4(2)).33C2, C42⋊C2.345C22, (C2×M4(2)).379C22, C8(C2×C4⋊C4), C4⋊C42(C2×C8), (C2×C4×C8)⋊40C2, C8(C2×C22⋊C4), C82(C2×C8⋊C4), (C2×C8)⋊40(C2×C4), C22⋊C42(C2×C8), C2.1(C2×C8○D4), (C2×C4⋊C4).83C4, C8(C2×C42⋊C2), (C2×C8⋊C4)⋊41C2, (C2×C8)3(C8⋊C4), C4⋊C4.242(C2×C4), C8⋊C42(C22×C8), (C2×C8)2(C2×M4(2)), C22⋊C4.85(C2×C4), (C2×C22⋊C4).54C4, C42⋊C2(C22×C8), (C2×C8)(C22×M4(2)), (C22×C8)(C2×M4(2)), (C2×C8)2(C42⋊C2), (C2×C8)(C82M4(2)), (C2×C4)(C82M4(2)), (C22×C4).381(C2×C4), (C2×C4).291(C22×C4), (C2×C42⋊C2).67C2, (C2×C8)(C2×C4⋊C4), (C2×C8)2(C2×C8⋊C4), (C22×C8)(C2×C8⋊C4), (C2×C8)(C2×C42⋊C2), (C22×C8)(C2×C42⋊C2), SmallGroup(128,1604)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C2×C82M4(2)
C1C2C22C2×C4C22×C4C23×C4C2×C42⋊C2 — C2×C82M4(2)
C1C2 — C2×C82M4(2)
C1C22×C8 — C2×C82M4(2)
C1C2C2C2×C4 — C2×C82M4(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 [×6], C2 [×4], C4 [×2], C4 [×6], C4 [×8], C22, C22 [×10], C22 [×12], C8 [×16], C2×C4 [×2], C2×C4 [×34], C2×C4 [×8], C23, C23 [×6], C23 [×4], C42 [×8], C22⋊C4 [×8], C4⋊C4 [×8], C2×C8 [×40], M4(2) [×16], C22×C4 [×2], C22×C4 [×16], C24, C4×C8 [×8], C8⋊C4 [×8], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×8], C22×C8 [×2], C22×C8 [×14], C2×M4(2) [×12], C23×C4, C2×C4×C8 [×2], C2×C8⋊C4 [×2], C82M4(2) [×8], C2×C42⋊C2, C23×C8, C22×M4(2), C2×C82M4(2)
Quotients: C1, C2 [×15], C4 [×24], C22 [×35], C2×C4 [×84], C23 [×15], C42 [×16], C22×C4 [×42], C24, C2×C42 [×12], C8○D4 [×4], C23×C4 [×3], C82M4(2) [×4], C22×C42, C2×C8○D4 [×2], 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 60)(10 61)(11 62)(12 63)(13 64)(14 57)(15 58)(16 59)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 42)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 41)
(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 40 5 62 20 36)(2 59 17 33 6 63 21 37)(3 60 18 34 7 64 22 38)(4 61 19 35 8 57 23 39)(9 26 43 53 13 30 47 49)(10 27 44 54 14 31 48 50)(11 28 45 55 15 32 41 51)(12 29 46 56 16 25 42 52)
(9 13)(10 14)(11 15)(12 16)(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)(9,60)(10,61)(11,62)(12,63)(13,64)(14,57)(15,58)(16,59)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,41), (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), (1,58,24,40,5,62,20,36)(2,59,17,33,6,63,21,37)(3,60,18,34,7,64,22,38)(4,61,19,35,8,57,23,39)(9,26,43,53,13,30,47,49)(10,27,44,54,14,31,48,50)(11,28,45,55,15,32,41,51)(12,29,46,56,16,25,42,52), (9,13)(10,14)(11,15)(12,16)(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)(9,60)(10,61)(11,62)(12,63)(13,64)(14,57)(15,58)(16,59)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,41), (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), (1,58,24,40,5,62,20,36)(2,59,17,33,6,63,21,37)(3,60,18,34,7,64,22,38)(4,61,19,35,8,57,23,39)(9,26,43,53,13,30,47,49)(10,27,44,54,14,31,48,50)(11,28,45,55,15,32,41,51)(12,29,46,56,16,25,42,52), (9,13)(10,14)(11,15)(12,16)(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),(9,60),(10,61),(11,62),(12,63),(13,64),(14,57),(15,58),(16,59),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(33,42),(34,43),(35,44),(36,45),(37,46),(38,47),(39,48),(40,41)], [(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)], [(1,58,24,40,5,62,20,36),(2,59,17,33,6,63,21,37),(3,60,18,34,7,64,22,38),(4,61,19,35,8,57,23,39),(9,26,43,53,13,30,47,49),(10,27,44,54,14,31,48,50),(11,28,45,55,15,32,41,51),(12,29,46,56,16,25,42,52)], [(9,13),(10,14),(11,15),(12,16),(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···2G2H2I2J2K4A···4H4I···4AB8A···8P8Q···8AN
order12···222224···44···48···88···8
size11···122221···12···21···12···2

80 irreducible representations

dim1111111111112
type+++++++
imageC1C2C2C2C2C2C2C4C4C4C4C4C8○D4
kernelC2×C82M4(2)C2×C4×C8C2×C8⋊C4C82M4(2)C2×C42⋊C2C23×C8C22×M4(2)C2×C22⋊C4C2×C4⋊C4C42⋊C2C22×C8C2×M4(2)C22
# reps1228111448161616

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

16000
01600
0010
0001
,
13000
01600
0080
0008
,
1000
01600
0090
00138
,
16000
01600
00113
00016
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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