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G = (C22×C8)⋊C2order 64 = 26

2nd semidirect product of C22×C8 and C2 acting faithfully

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

Aliases: (C22×C8)⋊2C2, (C2×D4).5C4, C4.68(C2×D4), (C2×Q8).5C4, C22⋊C812C2, C2.4(C8○D4), (C2×C4).118D4, C4.8(C22⋊C4), (C2×M4(2))⋊7C2, (C2×C8).57C22, C23.16(C2×C4), (C2×C4).146C23, C22.1(C22⋊C4), (C22×C4).29C22, C22.42(C22×C4), (C2×C4).42(C2×C4), (C2×C4○D4).1C2, C2.11(C2×C22⋊C4), SmallGroup(64,89)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — (C22×C8)⋊C2
C1C2C4C2×C4C22×C4C2×C4○D4 — (C22×C8)⋊C2
C1C22 — (C22×C8)⋊C2
C1C2×C4 — (C22×C8)⋊C2
C1C2C2C2×C4 — (C22×C8)⋊C2

Generators and relations for (C22×C8)⋊C2
 G = < a,b,c,d | a2=b2=c8=d2=1, ab=ba, ac=ca, dad=ac4, dcd=bc=cb, bd=db >

Subgroups: 121 in 79 conjugacy classes, 41 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×4], D4 [×6], Q8 [×2], C23, C23 [×2], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], C22⋊C8 [×4], C22×C8, C2×M4(2), C2×C4○D4, (C22×C8)⋊C2
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×C22⋊C4, C8○D4 [×2], (C22×C8)⋊C2

Character table of (C22×C8)⋊C2

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H8I8J8K8L
 size 1111224411112244222222224444
ρ11111111111111111111111111111    trivial
ρ21111-1-1-111111-1-1-11-11-1-111-111-11-1    linear of order 2
ρ31111-1-11-11111-1-11-11-111-1-11-11-11-1    linear of order 2
ρ4111111-1-1111111-1-1-1-1-1-1-1-1-1-11111    linear of order 2
ρ51111-1-1-111111-1-1-111-111-1-11-1-11-11    linear of order 2
ρ61111111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ7111111-1-1111111-1-111111111-1-1-1-1    linear of order 2
ρ81111-1-11-11111-1-11-1-11-1-111-11-11-11    linear of order 2
ρ9111111-11-1-1-1-1-1-11-1ii-i-i-i-iiiii-i-i    linear of order 4
ρ10111111-11-1-1-1-1-1-11-1-i-iiiii-i-i-i-iii    linear of order 4
ρ111111-1-1-1-1-1-1-1-11111-iiii-i-i-ii-iii-i    linear of order 4
ρ121111-1-1-1-1-1-1-1-11111i-i-i-iiii-ii-i-ii    linear of order 4
ρ131111-1-111-1-1-1-111-1-1-iiii-i-i-iii-i-ii    linear of order 4
ρ141111-1-111-1-1-1-111-1-1i-i-i-iiii-i-iii-i    linear of order 4
ρ151111111-1-1-1-1-1-1-1-11ii-i-i-i-iii-i-iii    linear of order 4
ρ161111111-1-1-1-1-1-1-1-11-i-iiiii-i-iii-i-i    linear of order 4
ρ172-2-22-2200-2-222-2200000000000000    orthogonal lifted from D4
ρ182-2-222-200-2-2222-200000000000000    orthogonal lifted from D4
ρ192-2-22-220022-2-22-200000000000000    orthogonal lifted from D4
ρ202-2-222-20022-2-2-2200000000000000    orthogonal lifted from D4
ρ2122-2-200002i-2i2i-2i0000830885008700000    complex lifted from C8○D4
ρ2222-2-200002i-2i2i-2i0000870858008300000    complex lifted from C8○D4
ρ232-22-200002i-2i-2i2i0000080087830850000    complex lifted from C8○D4
ρ2422-2-20000-2i2i-2i2i0000808387008500000    complex lifted from C8○D4
ρ252-22-20000-2i2i2i-2i0000083008580870000    complex lifted from C8○D4
ρ2622-2-20000-2i2i-2i2i0000850878300800000    complex lifted from C8○D4
ρ272-22-20000-2i2i2i-2i0000087008850830000    complex lifted from C8○D4
ρ282-22-200002i-2i-2i2i0000085008387080000    complex lifted from C8○D4

Smallest permutation representation of (C22×C8)⋊C2
On 32 points
Generators in S32
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)
(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)
(1 13)(2 18)(3 15)(4 20)(5 9)(6 22)(7 11)(8 24)(10 28)(12 30)(14 32)(16 26)(17 31)(19 25)(21 27)(23 29)

G:=sub<Sym(32)| (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20), (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), (1,13)(2,18)(3,15)(4,20)(5,9)(6,22)(7,11)(8,24)(10,28)(12,30)(14,32)(16,26)(17,31)(19,25)(21,27)(23,29)>;

G:=Group( (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20), (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), (1,13)(2,18)(3,15)(4,20)(5,9)(6,22)(7,11)(8,24)(10,28)(12,30)(14,32)(16,26)(17,31)(19,25)(21,27)(23,29) );

G=PermutationGroup([(1,15),(2,16),(3,9),(4,10),(5,11),(6,12),(7,13),(8,14),(17,29),(18,30),(19,31),(20,32),(21,25),(22,26),(23,27),(24,28)], [(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,21),(10,22),(11,23),(12,24),(13,17),(14,18),(15,19),(16,20)], [(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)], [(1,13),(2,18),(3,15),(4,20),(5,9),(6,22),(7,11),(8,24),(10,28),(12,30),(14,32),(16,26),(17,31),(19,25),(21,27),(23,29)])

(C22×C8)⋊C2 is a maximal subgroup of
C23.2C42  C23.3C42  (C22×C8)⋊C4  2+ 1+4.2C4  2+ 1+44C4  M4(2).40D4  C24.73(C2×C4)  D4○(C22⋊C8)  C42.261C23  C42.264C23  C42.265C23  C42.681C23  C42.266C23  M4(2)⋊22D4  M4(2)⋊23D4  C42.297C23  C42.298C23  C42.299C23  C42.300C23  C4○D4⋊D4  D4.(C2×D4)  (C2×Q8)⋊16D4  Q8.(C2×D4)  (C2×D4)⋊21D4  (C2×Q8)⋊17D4  (C2×D4).301D4  (C2×D4).302D4  (C2×D4).303D4  (C2×D4).304D4  C4.2+ 1+4  C4.142+ 1+4  C4.152+ 1+4  C4.162+ 1+4  C4.172+ 1+4  C4.182+ 1+4  C4.192+ 1+4
 (C2×C4p).D4: C23.M4(2)  C23.1M4(2)  (C2×D4).Q8  M4(2).43D4  M4(2).44D4  M4(2).24D4  C42.428D4  C42.107D4 ...
 C2p.(C8○D4): C42.260C23  C42.678C23  C42.694C23  C42.301C23  D6⋊C8⋊C2  C22⋊C8⋊D5  (C2×D4).7F5  (C2×D4).8F5 ...
(C22×C8)⋊C2 is a maximal quotient of
C23⋊C8⋊C2  C42.395D4  C42.396D4  C24.(C2×C4)  C24.45(C2×C4)  C42.372D4  C23.29C42  C42.379D4  C24.51(C2×C4)  C42.95D4  C24.53(C2×C4)  C23.22M4(2)  C232M4(2)  C42.325D4  C42.109D4  C42.327D4  C42.120D4
 D2p⋊C8⋊C2: C42.45D4  C42.373D4  C42.47D4  C42.400D4  C42.315D4  C42.305D4  C42.52D4  C42.53D4 ...
 C2p.(C8○D4): C42.46D4  C42.401D4  C42.316D4  C42.54D4  (C6×D4).11C4  C22⋊C8⋊D5  (C22×C8)⋊D5  C4.89(C2×D20) ...

Matrix representation of (C22×C8)⋊C2 in GL4(𝔽17) generated by

0100
1000
00160
00016
,
16000
01600
00160
00016
,
0900
9000
001515
00102
,
01300
4000
00160
0021
G:=sub<GL(4,GF(17))| [0,1,0,0,1,0,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[0,9,0,0,9,0,0,0,0,0,15,10,0,0,15,2],[0,4,0,0,13,0,0,0,0,0,16,2,0,0,0,1] >;

(C22×C8)⋊C2 in GAP, Magma, Sage, TeX

(C_2^2\times C_8)\rtimes C_2
% in TeX

G:=Group("(C2^2xC8):C2");
// GroupNames label

G:=SmallGroup(64,89);
// by ID

G=gap.SmallGroup(64,89);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,332,88]);
// Polycyclic

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

Export

Character table of (C22×C8)⋊C2 in TeX

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