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G = C2×C4.4D8order 128 = 27

Direct product of C2 and C4.4D8

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

Aliases: C2×C4.4D8, C42.352D4, C42.703C23, (C2×C4).92D8, C4.19(C2×D8), (C4×C8)⋊70C22, C4⋊Q861C22, C4⋊C4.81C23, C4.20(C2×SD16), (C2×C4).79SD16, C22.72(C2×D8), C2.10(C22×D8), (C2×C8).488C23, (C2×C4).326C24, (C2×D4).96C23, (C22×C4).608D4, C23.870(C2×D4), D4⋊C459C22, C4.18(C4.4D4), C2.16(C22×SD16), C22.86(C2×SD16), C41D4.143C22, (C22×C8).517C22, C22.586(C22×D4), (C2×C42).1121C22, (C22×C4).1548C23, C22.80(C4.4D4), (C22×D4).364C22, (C2×C4×C8)⋊20C2, (C2×C4⋊Q8)⋊34C2, C4.35(C2×C4○D4), (C2×C4).849(C2×D4), (C2×D4⋊C4)⋊18C2, (C2×C41D4).22C2, C2.37(C2×C4.4D4), (C2×C4).705(C4○D4), (C2×C4⋊C4).618C22, SmallGroup(128,1860)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×C4.4D8
C1C2C4C2×C4C22×C4C22×C8C2×C4×C8 — C2×C4.4D8
C1C2C2×C4 — C2×C4.4D8
C1C23C2×C42 — C2×C4.4D8
C1C2C2C2×C4 — C2×C4.4D8

Generators and relations for C2×C4.4D8
 G = < a,b,c,d | a2=b4=c8=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 628 in 264 conjugacy classes, 116 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×12], C4 [×4], C22, C22 [×6], C22 [×20], C8 [×4], C2×C4 [×2], C2×C4 [×16], C2×C4 [×8], D4 [×28], Q8 [×8], C23, C23 [×16], C42 [×4], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×2], C2×D4 [×4], C2×D4 [×26], C2×Q8 [×8], C24 [×2], C4×C8 [×4], D4⋊C4 [×16], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4, C41D4 [×4], C41D4 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C22×C8 [×2], C22×D4 [×2], C22×D4 [×2], C22×Q8, C2×C4×C8, C2×D4⋊C4 [×4], C4.4D8 [×8], C2×C41D4, C2×C4⋊Q8, C2×C4.4D8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], SD16 [×4], C2×D4 [×6], C4○D4 [×4], C24, C4.4D4 [×4], C2×D8 [×6], C2×SD16 [×6], C22×D4, C2×C4○D4 [×2], C4.4D8 [×4], C2×C4.4D4, C22×D8, C22×SD16, C2×C4.4D8

Smallest permutation representation of C2×C4.4D8
On 64 points
Generators in S64
(1 55)(2 56)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 41)(32 42)(33 62)(34 63)(35 64)(36 57)(37 58)(38 59)(39 60)(40 61)
(1 60 24 31)(2 61 17 32)(3 62 18 25)(4 63 19 26)(5 64 20 27)(6 57 21 28)(7 58 22 29)(8 59 23 30)(9 46 52 36)(10 47 53 37)(11 48 54 38)(12 41 55 39)(13 42 56 40)(14 43 49 33)(15 44 50 34)(16 45 51 35)
(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 23 24 8)(2 7 17 22)(3 21 18 6)(4 5 19 20)(9 14 52 49)(10 56 53 13)(11 12 54 55)(15 16 50 51)(25 28 62 57)(26 64 63 27)(29 32 58 61)(30 60 59 31)(33 36 43 46)(34 45 44 35)(37 40 47 42)(38 41 48 39)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4L4M4N4O4P8A···8P
order12···222224···444448···8
size11···188882···288882···2

44 irreducible representations

dim11111122222
type+++++++++
imageC1C2C2C2C2C2D4D4D8SD16C4○D4
kernelC2×C4.4D8C2×C4×C8C2×D4⋊C4C4.4D8C2×C41D4C2×C4⋊Q8C42C22×C4C2×C4C2×C4C2×C4
# reps11481122888

Matrix representation of C2×C4.4D8 in GL5(𝔽17)

160000
016000
001600
00010
00001
,
10000
00400
04000
00001
000160
,
160000
00100
01000
00055
000125
,
10000
00100
016000
000125
00055

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0,0,0,0,16,0,0,0,1,0],[16,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,5,12,0,0,0,5,5],[1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,12,5,0,0,0,5,5] >;

C2×C4.4D8 in GAP, Magma, Sage, TeX

C_2\times C_4._4D_8
% in TeX

G:=Group("C2xC4.4D8");
// GroupNames label

G:=SmallGroup(128,1860);
// by ID

G=gap.SmallGroup(128,1860);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,232,758,100,2804,172,4037,124]);
// Polycyclic

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

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