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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, 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···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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