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G = C4×2- 1+4order 128 = 27

Direct product of C4 and 2- 1+4

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

Aliases: C4×2- 1+4, C22.16C25, C42.537C23, C23.110C24, Q8(C4×D4), D4(C4×Q8), C2.12(C24×C4), C4.42(C23×C4), C4⋊C4.517C23, (C2×C4).162C24, D4.25(C22×C4), C22.6(C23×C4), Q8.26(C22×C4), (C4×D4).348C22, (C2×D4).500C23, (C2×Q8).483C23, (C4×Q8).318C22, C2.3(C2×2- 1+4), C22⋊C4.129C23, C2.3(C2.C25), (C2×C42).914C22, (C2×2- 1+4).8C2, (C22×C4).1296C23, (C22×Q8).483C22, C42(C23.32C23), C43(C23.33C23), C23.33C2335C2, C23.32C2321C2, C42⋊C2.336C22, C4⋊C4(C4×Q8), (C4×D4)(C4×Q8), (C2×D4)(C4×Q8), (C2×C4×Q8)⋊43C2, C22⋊C4(C4×Q8), C4○D411(C2×C4), (C4×C4○D4)⋊15C2, (C2×Q8)⋊30(C2×C4), (C2×C4).91(C22×C4), (C2×C4⋊C4).942C22, (C2×C4○D4).319C22, SmallGroup(128,2162)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C4×2- 1+4
C1C2C22C2×C4C22×C4C2×C42C2×C4×Q8 — C4×2- 1+4
C1C2 — C4×2- 1+4
C1C2×C4 — C4×2- 1+4
C1C22 — C4×2- 1+4

Generators and relations for C4×2- 1+4
 G = < a,b,c,d,e | a4=b4=c2=1, d2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 836 in 746 conjugacy classes, 686 normal (8 characteristic)
C1, C2 [×3], C2 [×10], C4 [×22], C4 [×15], C22, C22 [×10], C22 [×10], C2×C4, C2×C4 [×85], C2×C4 [×25], D4 [×40], Q8 [×40], C23 [×5], C42 [×40], C22⋊C4 [×20], C4⋊C4 [×40], C22×C4 [×35], C2×D4 [×10], C2×Q8 [×50], C4○D4 [×80], C2×C42 [×15], C2×C4⋊C4 [×15], C42⋊C2 [×30], C4×D4 [×40], C4×Q8 [×40], C22×Q8 [×5], C2×C4○D4 [×10], 2- 1+4 [×16], C2×C4×Q8 [×5], C4×C4○D4 [×10], C23.32C23 [×5], C23.33C23 [×10], C2×2- 1+4, C4×2- 1+4
Quotients: C1, C2 [×31], C4 [×16], C22 [×155], C2×C4 [×120], C23 [×155], C22×C4 [×140], C24 [×31], C23×C4 [×30], 2- 1+4 [×2], C25, C24×C4, C2×2- 1+4, C2.C25, C4×2- 1+4

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

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

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

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

68 conjugacy classes

class 1 2A2B2C2D···2M4A4B4C4D4E···4BB
order12222···244444···4
size11112···211112···2

68 irreducible representations

dim111111144
type++++++-
imageC1C2C2C2C2C2C42- 1+4C2.C25
kernelC4×2- 1+4C2×C4×Q8C4×C4○D4C23.32C23C23.33C23C2×2- 1+42- 1+4C4C2
# reps151051013222

Matrix representation of C4×2- 1+4 in GL5(𝔽5)

20000
01000
00100
00010
00001
,
40000
00020
00003
02000
00300
,
10000
00100
01000
00001
00010
,
40000
00001
00010
00400
04000
,
40000
00030
00003
03000
00300

G:=sub<GL(5,GF(5))| [2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,2,0,0,0,0,0,3,0,0],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0],[4,0,0,0,0,0,0,0,0,4,0,0,0,4,0,0,0,1,0,0,0,1,0,0,0],[4,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,3,0,0,0,0,0,3,0,0] >;

C4×2- 1+4 in GAP, Magma, Sage, TeX

C_4\times 2_-^{1+4}
% in TeX

G:=Group("C4xES-(2,2)");
// GroupNames label

G:=SmallGroup(128,2162);
// by ID

G=gap.SmallGroup(128,2162);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,232,387,184,1123,172]);
// Polycyclic

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

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