p-group, metabelian, nilpotent (class 2), monomial
Aliases: (C22xC8):2C2, (C2xD4).5C4, C4.68(C2xD4), (C2xQ8).5C4, C22:C8:12C2, C2.4(C8oD4), (C2xC4).118D4, C4.8(C22:C4), (C2xM4(2)):7C2, (C2xC8).57C22, C23.16(C2xC4), (C2xC4).146C23, C22.1(C22:C4), (C22xC4).29C22, C22.42(C22xC4), (C2xC4).42(C2xC4), (C2xC4oD4).1C2, C2.11(C2xC22:C4), SmallGroup(64,89)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C22xC8):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, C2, C4, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C2xC8, C2xC8, M4(2), C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C22:C8, C22xC8, C2xM4(2), C2xC4oD4, (C22xC8):C2
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22:C4, C22xC4, C2xD4, C2xC22:C4, C8oD4, (C22xC8):C2
Character table of (C22xC8):C2
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | |
size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | |
ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
ρ2 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | linear of order 2 |
ρ3 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
ρ5 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | linear of order 2 |
ρ9 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | i | i | -i | -i | -i | -i | i | i | i | i | -i | -i | linear of order 4 |
ρ10 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | -i | -i | i | i | i | i | -i | -i | -i | -i | i | i | linear of order 4 |
ρ11 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -i | i | i | i | -i | -i | -i | i | -i | i | i | -i | linear of order 4 |
ρ12 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | i | -i | -i | -i | i | i | i | -i | i | -i | -i | i | linear of order 4 |
ρ13 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -i | i | i | i | -i | -i | -i | i | i | -i | -i | i | linear of order 4 |
ρ14 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | i | -i | -i | -i | i | i | i | -i | -i | i | i | -i | linear of order 4 |
ρ15 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | i | i | -i | -i | -i | -i | i | i | -i | -i | i | i | linear of order 4 |
ρ16 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -i | -i | i | i | i | i | -i | -i | i | i | -i | -i | linear of order 4 |
ρ17 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | -2 | -2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ18 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | -2 | -2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ19 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ20 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 2ζ83 | 0 | 2ζ8 | 2ζ85 | 0 | 0 | 2ζ87 | 0 | 0 | 0 | 0 | 0 | complex lifted from C8oD4 |
ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 2ζ87 | 0 | 2ζ85 | 2ζ8 | 0 | 0 | 2ζ83 | 0 | 0 | 0 | 0 | 0 | complex lifted from C8oD4 |
ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 2ζ8 | 0 | 0 | 2ζ87 | 2ζ83 | 0 | 2ζ85 | 0 | 0 | 0 | 0 | complex lifted from C8oD4 |
ρ24 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 2ζ8 | 0 | 2ζ83 | 2ζ87 | 0 | 0 | 2ζ85 | 0 | 0 | 0 | 0 | 0 | complex lifted from C8oD4 |
ρ25 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 2ζ83 | 0 | 0 | 2ζ85 | 2ζ8 | 0 | 2ζ87 | 0 | 0 | 0 | 0 | complex lifted from C8oD4 |
ρ26 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 2ζ85 | 0 | 2ζ87 | 2ζ83 | 0 | 0 | 2ζ8 | 0 | 0 | 0 | 0 | 0 | complex lifted from C8oD4 |
ρ27 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 2ζ87 | 0 | 0 | 2ζ8 | 2ζ85 | 0 | 2ζ83 | 0 | 0 | 0 | 0 | complex lifted from C8oD4 |
ρ28 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 2ζ85 | 0 | 0 | 2ζ83 | 2ζ87 | 0 | 2ζ8 | 0 | 0 | 0 | 0 | complex lifted from C8oD4 |
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(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 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(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 12)(2 18)(3 14)(4 20)(5 16)(6 22)(7 10)(8 24)(9 28)(11 30)(13 32)(15 26)(17 31)(19 25)(21 27)(23 29)
G:=sub<Sym(32)| (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(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)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (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), (1,12)(2,18)(3,14)(4,20)(5,16)(6,22)(7,10)(8,24)(9,28)(11,30)(13,32)(15,26)(17,31)(19,25)(21,27)(23,29)>;
G:=Group( (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(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)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (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), (1,12)(2,18)(3,14)(4,20)(5,16)(6,22)(7,10)(8,24)(9,28)(11,30)(13,32)(15,26)(17,31)(19,25)(21,27)(23,29) );
G=PermutationGroup([[(1,14),(2,15),(3,16),(4,9),(5,10),(6,11),(7,12),(8,13),(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),(9,22),(10,23),(11,24),(12,17),(13,18),(14,19),(15,20),(16,21)], [(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)], [(1,12),(2,18),(3,14),(4,20),(5,16),(6,22),(7,10),(8,24),(9,28),(11,30),(13,32),(15,26),(17,31),(19,25),(21,27),(23,29)]])
(C22xC8):C2 is a maximal subgroup of
C23.2C42 C23.3C42 (C22xC8):C4 2+ 1+4.2C4 2+ 1+4:4C4 M4(2).40D4 C24.73(C2xC4) D4o(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 C4oD4:D4 D4.(C2xD4) (C2xQ8):16D4 Q8.(C2xD4) (C2xD4):21D4 (C2xQ8):17D4 (C2xD4).301D4 (C2xD4).302D4 (C2xD4).303D4 (C2xD4).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
(C2xC4p).D4: C23.M4(2) C23.1M4(2) (C2xD4).Q8 M4(2).43D4 M4(2).44D4 M4(2).24D4 C42.428D4 C42.107D4 ...
C2p.(C8oD4): C42.260C23 C42.678C23 C42.694C23 C42.301C23 D6:C8:C2 C22:C8:D5 (C2xD4).7F5 (C2xD4).8F5 ...
(C22xC8):C2 is a maximal quotient of
C23:C8:C2 C42.395D4 C42.396D4 C24.(C2xC4) C24.45(C2xC4) C42.372D4 C23.29C42 C42.379D4 C24.51(C2xC4) C42.95D4 C24.53(C2xC4) C23.22M4(2) C23:2M4(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.(C8oD4): C42.46D4 C42.401D4 C42.316D4 C42.54D4 (C6xD4).11C4 C22:C8:D5 (C22xC8):D5 C4.89(C2xD20) ...
Matrix representation of (C22xC8):C2 ►in GL4(F17) generated by
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 | 15 |
0 | 0 | 10 | 2 |
0 | 13 | 0 | 0 |
4 | 0 | 0 | 0 |
0 | 0 | 16 | 0 |
0 | 0 | 2 | 1 |
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] >;
(C22xC8):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
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