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G = C2×Q8⋊C4order 64 = 26

Direct product of C2 and Q8⋊C4

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

Aliases: C2×Q8⋊C4, C23.55D4, C22.5Q16, C22.11SD16, Q83(C2×C4), (C2×Q8)⋊5C4, C4.48(C2×D4), (C2×C4).70D4, C2.1(C2×Q16), C4.2(C22×C4), (C22×C8).5C2, C2.2(C2×SD16), C4⋊C4.42C22, (C2×C4).60C23, (C2×C8).58C22, C22.42(C2×D4), (C22×Q8).4C2, C4.13(C22⋊C4), (C2×Q8).40C22, C22.33(C22⋊C4), (C22×C4).108C22, (C2×C4⋊C4).12C2, (C2×C4).44(C2×C4), C2.18(C2×C22⋊C4), SmallGroup(64,96)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C2×Q8⋊C4
C1C2C22C2×C4C22×C4C22×Q8 — C2×Q8⋊C4
C1C2C4 — C2×Q8⋊C4
C1C23C22×C4 — C2×Q8⋊C4
C1C2C2C2×C4 — C2×Q8⋊C4

Generators and relations for C2×Q8⋊C4
 G = < a,b,c,d | a2=b4=d4=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b-1c >

Subgroups: 121 in 81 conjugacy classes, 49 normal (15 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×6], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], Q8 [×4], Q8 [×6], C23, C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×2], C22×C4, C22×C4 [×2], C2×Q8 [×6], C2×Q8 [×3], Q8⋊C4 [×4], C2×C4⋊C4, C22×C8, C22×Q8, C2×Q8⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], SD16 [×2], Q16 [×2], C22×C4, C2×D4 [×2], Q8⋊C4 [×4], C2×C22⋊C4, C2×SD16, C2×Q16, C2×Q8⋊C4

Character table of C2×Q8⋊C4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F8G8H
 size 1111111122224444444422222222
ρ11111111111111111111111111111    trivial
ρ21-11-1-11-1111-1-11-1-1111-1-1-11-11-11-11    linear of order 2
ρ31-11-1-11-1111-1-1-111-1-1-111-11-11-11-11    linear of order 2
ρ4111111111111-1-1-1-1-1-1-1-111111111    linear of order 2
ρ51-11-1-11-1111-1-11-11-1-11-111-11-11-11-1    linear of order 2
ρ611111111111111-1-1-111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ7111111111111-1-1111-1-11-1-1-1-1-1-1-1-1    linear of order 2
ρ81-11-1-11-1111-1-1-11-111-11-11-11-11-11-1    linear of order 2
ρ911-11-11-1-1-11-111-1i-ii-11-ii-ii-i-ii-ii    linear of order 4
ρ1011-11-11-1-1-11-111-1-ii-i-11i-ii-iii-ii-i    linear of order 4
ρ111-1-1-1111-1-111-1-1-1-i-ii11iiiii-i-i-i-i    linear of order 4
ρ121-1-1-1111-1-111-1-1-1ii-i11-i-i-i-i-iiiii    linear of order 4
ρ131-1-1-1111-1-111-111ii-i-1-1-iiiii-i-i-i-i    linear of order 4
ρ141-1-1-1111-1-111-111-i-ii-1-1i-i-i-i-iiiii    linear of order 4
ρ1511-11-11-1-1-11-11-11-ii-i1-1ii-ii-i-ii-ii    linear of order 4
ρ1611-11-11-1-1-11-11-11i-ii1-1-i-ii-iii-ii-i    linear of order 4
ρ1722222222-2-2-2-20000000000000000    orthogonal lifted from D4
ρ1822-22-22-2-22-22-20000000000000000    orthogonal lifted from D4
ρ192-2-2-2222-22-2-220000000000000000    orthogonal lifted from D4
ρ202-22-2-22-22-2-2220000000000000000    orthogonal lifted from D4
ρ2122-2-22-2-2200000000000022-2-2-2-222    symplectic lifted from Q16, Schur index 2
ρ2222-2-22-2-22000000000000-2-22222-2-2    symplectic lifted from Q16, Schur index 2
ρ232-2-22-2-2220000000000002-2-22-222-2    symplectic lifted from Q16, Schur index 2
ρ242-2-22-2-222000000000000-222-22-2-22    symplectic lifted from Q16, Schur index 2
ρ252-2222-2-2-2000000000000--2--2-2-2--2--2-2-2    complex lifted from SD16
ρ26222-2-2-22-2000000000000--2-2-2--2--2-2-2--2    complex lifted from SD16
ρ272-2222-2-2-2000000000000-2-2--2--2-2-2--2--2    complex lifted from SD16
ρ28222-2-2-22-2000000000000-2--2--2-2-2--2--2-2    complex lifted from SD16

Smallest permutation representation of C2×Q8⋊C4
Regular action on 64 points
Generators in S64
(1 20)(2 17)(3 18)(4 19)(5 56)(6 53)(7 54)(8 55)(9 14)(10 15)(11 16)(12 13)(21 33)(22 34)(23 35)(24 36)(25 29)(26 30)(27 31)(28 32)(37 50)(38 51)(39 52)(40 49)(41 45)(42 46)(43 47)(44 48)(57 62)(58 63)(59 64)(60 61)
(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 34 3 36)(2 33 4 35)(5 37 7 39)(6 40 8 38)(9 30 11 32)(10 29 12 31)(13 27 15 25)(14 26 16 28)(17 21 19 23)(18 24 20 22)(41 62 43 64)(42 61 44 63)(45 57 47 59)(46 60 48 58)(49 55 51 53)(50 54 52 56)
(1 49 11 46)(2 52 12 45)(3 51 9 48)(4 50 10 47)(5 28 62 22)(6 27 63 21)(7 26 64 24)(8 25 61 23)(13 41 17 39)(14 44 18 38)(15 43 19 37)(16 42 20 40)(29 60 35 55)(30 59 36 54)(31 58 33 53)(32 57 34 56)

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

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

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

C2×Q8⋊C4 is a maximal subgroup of
Q8⋊C42  C24.155D4  C42.99D4  C42.101D4  Q8⋊(C4⋊C4)  C24.160D4  (C2×SD16)⋊15C4  C24.135D4  C24.75D4  C4.68(C4×D4)  C2.(C88D4)  C2.(C8⋊D4)  C42.431D4  C42.433D4  C42.110D4  C42.111D4  (C2×C4)⋊9SD16  (C2×C4)⋊6Q16  (C2×Q16)⋊10C4  C8⋊(C22⋊C4)  C42.117D4  (C2×C4)⋊3SD16  (C2×C4)⋊2Q16  (C2×Q8)⋊Q8  C24.86D4  C4⋊C4.95D4  (C2×C4)⋊3Q16  (C2×Q8).8Q8  (C2×C4).19Q16  (C2×Q8).109D4  (C2×C8).60D4  (C2×C8).170D4  (C2×C8).171D4  2- 1+44C4  C2×C4×SD16  C2×C4×Q16  C42.276C23  (C2×Q8)⋊17D4  C42.19C23  M4(2)⋊17D4  C42.21C23  (C2×D4).302D4  C42.367C23  D48SD16  D45Q16  C42.465C23  C42.466C23  C42.43C23  C42.47C23  C42.51C23  C42.55C23
 C2.(D4.pD4): C24.65D4  Q8⋊C4⋊C4  C24.73D4  (C2×SD16)⋊14C4  (C2×C4)⋊9Q16  M4(2).49D4  C2.(C4×Q16)  C4.10D43C4 ...
C2×Q8⋊C4 is a maximal quotient of
C42.46D4  Q8⋊M4(2)  C42.316D4  C42.404D4  C42.62D4  C24.61D4  C42.410D4  C42.415D4  C42.416D4  C42.79D4  C24.155D4  C42.99D4  C24.157D4  C24.160D4  C24.135D4  C42.431D4  C42.117D4  C42.121D4  C42.436D4

Matrix representation of C2×Q8⋊C4 in GL4(𝔽17) generated by

1000
01600
00160
00016
,
1000
0100
00115
00116
,
16000
0100
0049
00013
,
13000
01600
00107
0057
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,15,16],[16,0,0,0,0,1,0,0,0,0,4,0,0,0,9,13],[13,0,0,0,0,16,0,0,0,0,10,5,0,0,7,7] >;

C2×Q8⋊C4 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes C_4
% in TeX

G:=Group("C2xQ8:C4");
// GroupNames label

G:=SmallGroup(64,96);
// by ID

G=gap.SmallGroup(64,96);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,199,963,489,117]);
// Polycyclic

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

Export

Character table of C2×Q8⋊C4 in TeX

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