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G = Q8×C22⋊C4order 128 = 27

Direct product of Q8 and C22⋊C4

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

Aliases: Q8×C22⋊C4, C24.550C23, C23.222C24, C22.422- 1+4, C2.1(D4×Q8), C223(C4×Q8), (C2×Q8).258D4, (C22×Q8)⋊18C4, (Q8×C23).6C2, C2.1(Q85D4), C23.111(C2×Q8), C23.321(C4○D4), C22.36(C22×Q8), C22.113(C23×C4), (C23×C4).296C22, (C2×C42).425C22, C23.214(C22×C4), C22.101(C22×D4), C23.7Q8.28C2, (C22×C4).1243C23, (C22×Q8).507C22, C23.67C2319C2, C2.C42.56C22, C2.13(C23.32C23), (C2×C4×Q8)⋊6C2, C2.14(C2×C4×Q8), (C2×C4)⋊10(C2×Q8), C4.26(C2×C22⋊C4), (C2×C4).1067(C2×D4), (C4×C22⋊C4).24C2, (C2×Q8).194(C2×C4), (C2×C4⋊C4).816C22, (C2×C4).228(C22×C4), (C22×C4).306(C2×C4), C22.107(C2×C4○D4), C2.18(C22×C22⋊C4), (C2×C22⋊C4).434C22, (C2×Q8)(C2×C22⋊C4), SmallGroup(128,1072)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — Q8×C22⋊C4
C1C2C22C23C22×C4C23×C4Q8×C23 — Q8×C22⋊C4
C1C22 — Q8×C22⋊C4
C1C23 — Q8×C22⋊C4
C1C23 — Q8×C22⋊C4

Generators and relations for Q8×C22⋊C4
 G = < a,b,c,d,e | a4=c2=d2=e4=1, b2=a2, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >

Subgroups: 636 in 410 conjugacy classes, 196 normal (13 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×12], C4 [×16], C22 [×3], C22 [×8], C22 [×12], C2×C4 [×28], C2×C4 [×52], Q8 [×16], Q8 [×24], C23, C23 [×6], C23 [×4], C42 [×12], C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×12], C22×C4 [×26], C22×C4 [×12], C2×Q8 [×24], C2×Q8 [×44], C24, C2.C42 [×12], C2×C42 [×6], C2×C22⋊C4, C2×C22⋊C4 [×3], C2×C4⋊C4 [×6], C4×Q8 [×8], C23×C4 [×3], C22×Q8, C22×Q8 [×11], C22×Q8 [×8], C4×C22⋊C4 [×3], C23.7Q8 [×3], C23.67C23 [×6], C2×C4×Q8 [×2], Q8×C23, Q8×C22⋊C4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], Q8 [×4], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C2×Q8 [×6], C4○D4 [×2], C24, C2×C22⋊C4 [×12], C4×Q8 [×4], C23×C4, C22×D4 [×2], C22×Q8, C2×C4○D4, 2- 1+4 [×2], C22×C22⋊C4, C2×C4×Q8, C23.32C23, Q85D4 [×2], D4×Q8 [×2], Q8×C22⋊C4

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

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

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

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

50 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4T4U···4AL
order12···222224···44···4
size11···122222···24···4

50 irreducible representations

dim11111112224
type++++++-+-
imageC1C2C2C2C2C2C4Q8D4C4○D42- 1+4
kernelQ8×C22⋊C4C4×C22⋊C4C23.7Q8C23.67C23C2×C4×Q8Q8×C23C22×Q8C22⋊C4C2×Q8C23C22
# reps133621164842

Matrix representation of Q8×C22⋊C4 in GL5(𝔽5)

10000
03000
00200
00040
00004
,
40000
00100
04000
00010
00001
,
40000
04000
00400
00010
00004
,
10000
01000
00100
00040
00004
,
20000
03000
00300
00001
00010

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

Q8×C22⋊C4 in GAP, Magma, Sage, TeX

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

G:=Group("Q8xC2^2:C4");
// GroupNames label

G:=SmallGroup(128,1072);
// by ID

G=gap.SmallGroup(128,1072);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,346,80]);
// Polycyclic

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

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