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G = C4×C22⋊Q8order 128 = 27

Direct product of C4 and C22⋊Q8

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

Aliases: C4×C22⋊Q8, C42.438D4, C23.184C24, C24.538C23, C221(C4×Q8), C4.128(C4×D4), (C22×C4)⋊20Q8, C23.90(C2×Q8), C22.75(C23×C4), (C22×C42).25C2, C45(C23.7Q8), C45(C23.8Q8), C22.80(C22×D4), C23.219(C4○D4), C22.27(C22×Q8), (C23×C4).680C22, C23.121(C22×C4), (C2×C42).403C22, (C22×C4).748C23, C23.8Q8.73C2, C23.7Q8.83C2, C2.8(C22.19C24), (C22×Q8).392C22, C43(C23.67C23), C45(C23.63C23), C45(C23.65C23), C23.67C23113C2, C23.63C23211C2, C23.65C23175C2, C2.C42.517C22, C2.4(C23.37C23), C2.8(C23.36C23), (C2×C4×Q8)⋊2C2, C2.8(C2×C4×Q8), (C4×C4⋊C4)⋊21C2, C2.16(C2×C4×D4), C4⋊C426(C2×C4), (C2×Q8)⋊22(C2×C4), C2.13(C4×C4○D4), C2.6(C2×C22⋊Q8), (C2×C4).353(C2×Q8), (C2×C4).1391(C2×D4), (C4×C22⋊C4).19C2, C22⋊C4.29(C2×C4), (C2×C4).20(C22×C4), C22.76(C2×C4○D4), (C2×C22⋊Q8).62C2, (C2×C4).638(C4○D4), (C2×C4⋊C4).798C22, (C22×C4).412(C2×C4), (C2×C4)3(C23.7Q8), (C2×C22⋊C4).479C22, (C2×C4)3(C23.65C23), (C2×C4)(C2×C22⋊Q8), SmallGroup(128,1034)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C4×C22⋊Q8
C1C2C22C23C22×C4C23×C4C22×C42 — C4×C22⋊Q8
C1C22 — C4×C22⋊Q8
C1C22×C4 — C4×C22⋊Q8
C1C23 — C4×C22⋊Q8

Generators and relations for C4×C22⋊Q8
 G = < a,b,c,d,e | a4=b2=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 492 in 328 conjugacy classes, 172 normal (42 characteristic)
C1, C2 [×7], C2 [×4], C4 [×8], C4 [×20], C22 [×7], C22 [×4], C22 [×12], C2×C4 [×28], C2×C4 [×52], Q8 [×8], C23, C23 [×6], C23 [×4], C42 [×4], C42 [×14], C22⋊C4 [×8], C22⋊C4 [×4], C4⋊C4 [×12], C4⋊C4 [×12], C22×C4 [×6], C22×C4 [×16], C22×C4 [×14], C2×Q8 [×4], C2×Q8 [×4], C24, C2.C42 [×10], C2×C42 [×4], C2×C42 [×4], C2×C42 [×4], C2×C22⋊C4 [×4], C2×C4⋊C4 [×3], C2×C4⋊C4 [×6], C4×Q8 [×4], C22⋊Q8 [×8], C23×C4 [×3], C22×Q8, C4×C22⋊C4 [×2], C4×C4⋊C4, C4×C4⋊C4 [×2], C23.7Q8, C23.8Q8 [×2], C23.63C23 [×2], C23.65C23, C23.67C23, C22×C42, C2×C4×Q8, C2×C22⋊Q8, C4×C22⋊Q8
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×8], C24, C4×D4 [×4], C4×Q8 [×4], C22⋊Q8 [×4], C23×C4, C22×D4, C22×Q8, C2×C4○D4 [×4], C2×C4×D4, C2×C4×Q8, C4×C4○D4, C2×C22⋊Q8, C22.19C24, C23.36C23, C23.37C23, C4×C22⋊Q8

Smallest permutation representation of C4×C22⋊Q8
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 22)(2 23)(3 24)(4 21)(5 64)(6 61)(7 62)(8 63)(9 15)(10 16)(11 13)(12 14)(17 25)(18 26)(19 27)(20 28)(29 35)(30 36)(31 33)(32 34)(37 43)(38 44)(39 41)(40 42)(45 59)(46 60)(47 57)(48 58)(49 55)(50 56)(51 53)(52 54)
(1 26)(2 27)(3 28)(4 25)(5 12)(6 9)(7 10)(8 11)(13 63)(14 64)(15 61)(16 62)(17 21)(18 22)(19 23)(20 24)(29 37)(30 38)(31 39)(32 40)(33 41)(34 42)(35 43)(36 44)(45 53)(46 54)(47 55)(48 56)(49 57)(50 58)(51 59)(52 60)
(1 42 22 40)(2 43 23 37)(3 44 24 38)(4 41 21 39)(5 60 14 54)(6 57 15 55)(7 58 16 56)(8 59 13 53)(9 49 61 47)(10 50 62 48)(11 51 63 45)(12 52 64 46)(17 31 25 33)(18 32 26 34)(19 29 27 35)(20 30 28 36)
(1 45 22 51)(2 46 23 52)(3 47 24 49)(4 48 21 50)(5 35 14 29)(6 36 15 30)(7 33 16 31)(8 34 13 32)(9 44 61 38)(10 41 62 39)(11 42 63 40)(12 43 64 37)(17 58 25 56)(18 59 26 53)(19 60 27 54)(20 57 28 55)

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

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

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

56 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4AB4AC···4AR
order12···222224···44···44···4
size11···122221···12···24···4

56 irreducible representations

dim1111111111112222
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C4D4Q8C4○D4C4○D4
kernelC4×C22⋊Q8C4×C22⋊C4C4×C4⋊C4C23.7Q8C23.8Q8C23.63C23C23.65C23C23.67C23C22×C42C2×C4×Q8C2×C22⋊Q8C22⋊Q8C42C22×C4C2×C4C23
# reps123122111111644124

Matrix representation of C4×C22⋊Q8 in GL6(𝔽5)

300000
030000
001000
000100
000040
000004
,
400000
040000
004000
000400
000040
000001
,
100000
010000
001000
000100
000040
000004
,
200000
030000
003000
000200
000030
000002
,
030000
300000
000100
004000
000001
000040

G:=sub<GL(6,GF(5))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,2],[0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0] >;

C4×C22⋊Q8 in GAP, Magma, Sage, TeX

C_4\times C_2^2\rtimes Q_8
% in TeX

G:=Group("C4xC2^2:Q8");
// GroupNames label

G:=SmallGroup(128,1034);
// by ID

G=gap.SmallGroup(128,1034);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,304]);
// Polycyclic

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

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