p-group, metabelian, nilpotent (class 2), monomial
Aliases: C22.47C24, C42.48C22, C23.19C23, C2.162+ 1+4, (C4×D4)⋊20C2, C4⋊D4⋊15C2, C42.C2⋊9C2, C42⋊2C2⋊6C2, C4.36(C4○D4), C4⋊C4.36C22, (C2×C4).57C23, C42⋊C2⋊16C2, (C2×D4).35C22, C22.11(C4○D4), C22.D4⋊11C2, C22⋊C4.22C22, (C22×C4).14C22, (C2×C4⋊C4)⋊22C2, C2.26(C2×C4○D4), SmallGroup(64,234)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22.47C24
G = < a,b,c,d,e,f | a2=b2=c2=e2=1, d2=ba=ab, f2=a, dcd-1=ac=ca, fdf-1=ad=da, ae=ea, af=fa, ece=bc=cb, bd=db, be=eb, bf=fb, cf=fc, de=ed, ef=fe >
Subgroups: 181 in 119 conjugacy classes, 75 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4⋊D4, C4⋊D4, C22.D4, C42.C2, C42⋊2C2, C22.47C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, C22.47C24
Character table of C22.47C24
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 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 |
| ρ10 | 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 |
| ρ11 | 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 |
| ρ12 | 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 |
| ρ13 | 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 |
| ρ14 | 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 |
| ρ15 | 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 |
| ρ16 | 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 |
| ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | -2i | 2i | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ18 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 2i | 0 | 2i | -2i | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ19 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 2i | -2i | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ20 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 2i | 0 | -2i | -2i | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 2i | -2i | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | -2i | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | -2i | 2i | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | -2i | 0 | 2i | 2i | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
►On 32 points
(1 5)(2 6)(3 7)(4 8)(9 31)(10 32)(11 29)(12 30)(13 23)(14 24)(15 21)(16 22)(17 27)(18 28)(19 25)(20 26)
(1 7)(2 8)(3 5)(4 6)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)(19 27)(20 28)
(1 30)(2 9)(3 32)(4 11)(5 12)(6 31)(7 10)(8 29)(13 26)(14 17)(15 28)(16 19)(18 21)(20 23)(22 25)(24 27)
(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 31)(2 32)(3 29)(4 30)(5 9)(6 10)(7 11)(8 12)(13 17)(14 18)(15 19)(16 20)(21 25)(22 26)(23 27)(24 28)
(1 21 5 15)(2 16 6 22)(3 23 7 13)(4 14 8 24)(9 19 31 25)(10 26 32 20)(11 17 29 27)(12 28 30 18)
G:=sub<Sym(32)| (1,5)(2,6)(3,7)(4,8)(9,31)(10,32)(11,29)(12,30)(13,23)(14,24)(15,21)(16,22)(17,27)(18,28)(19,25)(20,26), (1,7)(2,8)(3,5)(4,6)(9,29)(10,30)(11,31)(12,32)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28), (1,30)(2,9)(3,32)(4,11)(5,12)(6,31)(7,10)(8,29)(13,26)(14,17)(15,28)(16,19)(18,21)(20,23)(22,25)(24,27), (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), (1,31)(2,32)(3,29)(4,30)(5,9)(6,10)(7,11)(8,12)(13,17)(14,18)(15,19)(16,20)(21,25)(22,26)(23,27)(24,28), (1,21,5,15)(2,16,6,22)(3,23,7,13)(4,14,8,24)(9,19,31,25)(10,26,32,20)(11,17,29,27)(12,28,30,18)>;
G:=Group( (1,5)(2,6)(3,7)(4,8)(9,31)(10,32)(11,29)(12,30)(13,23)(14,24)(15,21)(16,22)(17,27)(18,28)(19,25)(20,26), (1,7)(2,8)(3,5)(4,6)(9,29)(10,30)(11,31)(12,32)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28), (1,30)(2,9)(3,32)(4,11)(5,12)(6,31)(7,10)(8,29)(13,26)(14,17)(15,28)(16,19)(18,21)(20,23)(22,25)(24,27), (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), (1,31)(2,32)(3,29)(4,30)(5,9)(6,10)(7,11)(8,12)(13,17)(14,18)(15,19)(16,20)(21,25)(22,26)(23,27)(24,28), (1,21,5,15)(2,16,6,22)(3,23,7,13)(4,14,8,24)(9,19,31,25)(10,26,32,20)(11,17,29,27)(12,28,30,18) );
G=PermutationGroup([[(1,5),(2,6),(3,7),(4,8),(9,31),(10,32),(11,29),(12,30),(13,23),(14,24),(15,21),(16,22),(17,27),(18,28),(19,25),(20,26)], [(1,7),(2,8),(3,5),(4,6),(9,29),(10,30),(11,31),(12,32),(13,21),(14,22),(15,23),(16,24),(17,25),(18,26),(19,27),(20,28)], [(1,30),(2,9),(3,32),(4,11),(5,12),(6,31),(7,10),(8,29),(13,26),(14,17),(15,28),(16,19),(18,21),(20,23),(22,25),(24,27)], [(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)], [(1,31),(2,32),(3,29),(4,30),(5,9),(6,10),(7,11),(8,12),(13,17),(14,18),(15,19),(16,20),(21,25),(22,26),(23,27),(24,28)], [(1,21,5,15),(2,16,6,22),(3,23,7,13),(4,14,8,24),(9,19,31,25),(10,26,32,20),(11,17,29,27),(12,28,30,18)]])
C22.47C24 is a maximal subgroup of
C42.486C23 C42.488C23 C42.59C23 C42.60C23 C42.61C23 C42.64C23 C42.492C23 C42.496C23 C22.64C25 C22.69C25 C22.81C25 C22.82C25 C22.83C25 C22.101C25 C22.102C25 C22.104C25 C22.105C25 C22.110C25 C22.113C25 C22.122C25 C22.123C25 C22.142C25 C22.148C25 C22.149C25 C22.154C25 C22.155C25 C22.156C25
C2p.2+ 1+4: C42.462C23 C42.466C23 C42.42C23 C42.44C23 C42.53C23 C42.55C23 C42.471C23 C42.475C23 ...
C22.47C24 is a maximal quotient of
C23.227C24 C23.229C24 C23.234C24 C23.235C24 C24.212C23 C24.215C23 C24.217C23 C24.218C23 C23.252C24 C23.255C24 C24.223C23 C24.249C23 C23.316C24 C24.254C23 C23.322C24 C24.269C23 C23.344C24 C23.356C24 C24.278C23 C23.364C24 C24.286C23 C23.367C24 C23.368C24 C24.289C23 C24.293C23 C24.573C23 C23.385C24 C24.300C23 C24.304C23 C23.395C24 C23.397C24 C23.400C24 C23.404C24 C23.407C24 C23.409C24 C23.410C24 C23.412C24 C23.413C24 C24.309C23 C23.416C24 C23.418C24 C23.422C24 C23.425C24 C23.426C24 C23.429C24 C23.430C24 C23.431C24 C42⋊17D4 C23.443C24 C24.327C23 C24.331C23 C24.584C23 C42.36Q8 C23.473C24 C24.340C23 C24.341C23 C23.478C24 C23.479C24 C23.485C24 C24.345C23 C23.490C24 C23.491C24 C23.493C24 C23.494C24 C24.347C23 C23.496C24 C24.348C23 C42⋊22D4 C23.500C24 C42⋊23D4 C23.502C24 C42⋊24D4 C42.38Q8 C23.508C24 C24.395C23 C23.591C24 C24.407C23 C23.603C24 C23.608C24 C24.413C23 C23.618C24 C24.427C23 C23.641C24 C24.432C23 C23.647C24 C23.649C24 C24.435C23 C24.437C23 C23.656C24 C24.438C23 C24.440C23 C24.443C23 C23.666C24 C24.445C23 C23.672C24 C23.676C24 C23.677C24 C23.678C24 C23.679C24 C24.448C23 C23.681C24 C23.682C24 C23.683C24 C23.686C24 C23.687C24 C24.454C23 C23.693C24 C23.695C24 C23.696C24 C23.697C24 C23.700C24 C23.701C24 C23.702C24 C23.703C24
C42.D2p: C42.172D4 C42.175D4 C42.95D6 C42.104D6 C42.113D6 C42.119D6 C42.153D6 C42.163D6 ...
C4⋊C4.D2p: C24.268C23 C24.569C23 C23.360C24 C23.390C24 C23.458C24 C23.611C24 C23.625C24 C6.112+ 1+4 ...
Matrix representation of C22.47C24 ►in GL4(𝔽5) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 2 | 0 | 0 |
| 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 2 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 2 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 4 | 0 |
G:=sub<GL(4,GF(5))| [1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[0,3,0,0,2,0,0,0,0,0,0,2,0,0,3,0],[2,0,0,0,0,2,0,0,0,0,0,2,0,0,2,0],[0,1,0,0,1,0,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,0,4,0,0,1,0] >;
C22.47C24 in GAP, Magma, Sage, TeX
C_2^2._{47}C_2^4 % in TeX
G:=Group("C2^2.47C2^4"); // GroupNames label
G:=SmallGroup(64,234);
// by ID
G=gap.SmallGroup(64,234);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,217,295,650,297,69]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=e^2=1,d^2=b*a=a*b,f^2=a,d*c*d^-1=a*c=c*a,f*d*f^-1=a*d=d*a,a*e=e*a,a*f=f*a,e*c*e=b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*f=f*c,d*e=e*d,e*f=f*e>;
// generators/relations
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