direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C22xM4(2), C8:4C23, C24.5C4, C4.15C24, C4o(C2xM4(2)), (C2xC4)oM4(2), (C22xC8):12C2, (C2xC8):15C22, (C22xC4).18C4, C23.35(C2xC4), C4.31(C22xC4), C2.10(C23xC4), (C23xC4).11C2, (C2xC4).161C23, C22.27(C22xC4), (C22xC4).127C22, (C2xC4).77(C2xC4), (C2xC4)o(C2xM4(2)), SmallGroup(64,247)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22xM4(2)
G = < a,b,c,d | a2=b2=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c5 >
Subgroups: 169 in 149 conjugacy classes, 129 normal (9 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2xC4, C23, C23, C23, C2xC8, M4(2), C22xC4, C22xC4, C24, C22xC8, C2xM4(2), C23xC4, C22xM4(2)
Quotients: C1, C2, C4, C22, C2xC4, C23, M4(2), C22xC4, C24, C2xM4(2), C23xC4, C22xM4(2)
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)(17 30)(18 31)(19 32)(20 25)(21 26)(22 27)(23 28)(24 29)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(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 28)(2 25)(3 30)(4 27)(5 32)(6 29)(7 26)(8 31)(9 18)(10 23)(11 20)(12 17)(13 22)(14 19)(15 24)(16 21)
G:=sub<Sym(32)| (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,30)(18,31)(19,32)(20,25)(21,26)(22,27)(23,28)(24,29), (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(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), (1,28)(2,25)(3,30)(4,27)(5,32)(6,29)(7,26)(8,31)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21)>;
G:=Group( (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,30)(18,31)(19,32)(20,25)(21,26)(22,27)(23,28)(24,29), (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(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), (1,28)(2,25)(3,30)(4,27)(5,32)(6,29)(7,26)(8,31)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21) );
G=PermutationGroup([[(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,9),(17,30),(18,31),(19,32),(20,25),(21,26),(22,27),(23,28),(24,29)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,25),(7,26),(8,27),(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)], [(1,28),(2,25),(3,30),(4,27),(5,32),(6,29),(7,26),(8,31),(9,18),(10,23),(11,20),(12,17),(13,22),(14,19),(15,24),(16,21)]])
C22xM4(2) is a maximal subgroup of
C23.28C42 C23.29C42 C24.63D4 C24.152D4 C24.7Q8 C23.15C42 C23.17C42 C24.67D4 C24.9Q8 (C2xC8).195D4 C24.10Q8 C23:2M4(2) C24.72D4 C24.75D4 C24.76D4 M4(2):20D4 M4(2).45D4 M4(2)o2M4(2) C24.73(C2xC4) C24.98D4 C42.257C23 C24.100D4 C42.265C23 M4(2):22D4 C24.110D4 M4(2):14D4 M4(2):15D4
C22xM4(2) is a maximal quotient of
C42.677C23 C42.290C23 D4:6M4(2) Q8:6M4(2) C23:3M4(2) D4:7M4(2) C42.693C23 C42.302C23 Q8.4M4(2) C42.698C23 D4:8M4(2) Q8:7M4(2)
40 conjugacy classes
class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 8A | ··· | 8P |
order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
40 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
type | + | + | + | + | |||
image | C1 | C2 | C2 | C2 | C4 | C4 | M4(2) |
kernel | C22xM4(2) | C22xC8 | C2xM4(2) | C23xC4 | C22xC4 | C24 | C22 |
# reps | 1 | 2 | 12 | 1 | 14 | 2 | 8 |
Matrix representation of C22xM4(2) ►in GL4(F17) generated by
16 | 0 | 0 | 0 |
0 | 1 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
1 | 0 | 0 | 0 |
0 | 16 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
13 | 0 | 0 | 0 |
0 | 16 | 0 | 0 |
0 | 0 | 13 | 2 |
0 | 0 | 11 | 4 |
16 | 0 | 0 | 0 |
0 | 1 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 4 | 16 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[13,0,0,0,0,16,0,0,0,0,13,11,0,0,2,4],[16,0,0,0,0,1,0,0,0,0,1,4,0,0,0,16] >;
C22xM4(2) in GAP, Magma, Sage, TeX
C_2^2\times M_4(2)
% in TeX
G:=Group("C2^2xM4(2)");
// GroupNames label
G:=SmallGroup(64,247);
// by ID
G=gap.SmallGroup(64,247);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,-2,96,409,88]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^5>;
// generators/relations