direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C22×C4⋊C4, C23.59D4, C23.11Q8, C22.6C24, C24.36C22, C23.69C23, (C22×C4)⋊9C4, C4⋊2(C22×C4), (C23×C4).8C2, C2.2(C23×C4), C2.2(C22×D4), C2.1(C22×Q8), C23.40(C2×C4), (C2×C4).48C23, C22.58(C2×D4), C22.16(C2×Q8), C22.25(C22×C4), (C22×C4).97C22, (C2×C4)⋊10(C2×C4), SmallGroup(64,194)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22×C4⋊C4
G = < a,b,c,d | a2=b2=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 249 in 209 conjugacy classes, 169 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C4⋊C4, C22×C4, C22×C4, C24, C2×C4⋊C4, C23×C4, C23×C4, C22×C4⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C24, C2×C4⋊C4, C23×C4, C22×D4, C22×Q8, C22×C4⋊C4
►Regular action on 64 points
(1 15)(2 16)(3 13)(4 14)(5 9)(6 10)(7 11)(8 12)(17 41)(18 42)(19 43)(20 44)(21 31)(22 32)(23 29)(24 30)(25 56)(26 53)(27 54)(28 55)(33 61)(34 62)(35 63)(36 64)(37 51)(38 52)(39 49)(40 50)(45 57)(46 58)(47 59)(48 60)
(1 44)(2 41)(3 42)(4 43)(5 22)(6 23)(7 24)(8 21)(9 32)(10 29)(11 30)(12 31)(13 18)(14 19)(15 20)(16 17)(25 61)(26 62)(27 63)(28 64)(33 56)(34 53)(35 54)(36 55)(37 60)(38 57)(39 58)(40 59)(45 52)(46 49)(47 50)(48 51)
(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 54 12 57)(2 53 9 60)(3 56 10 59)(4 55 11 58)(5 48 16 26)(6 47 13 25)(7 46 14 28)(8 45 15 27)(17 62 22 51)(18 61 23 50)(19 64 24 49)(20 63 21 52)(29 40 42 33)(30 39 43 36)(31 38 44 35)(32 37 41 34)
G:=sub<Sym(64)| (1,15)(2,16)(3,13)(4,14)(5,9)(6,10)(7,11)(8,12)(17,41)(18,42)(19,43)(20,44)(21,31)(22,32)(23,29)(24,30)(25,56)(26,53)(27,54)(28,55)(33,61)(34,62)(35,63)(36,64)(37,51)(38,52)(39,49)(40,50)(45,57)(46,58)(47,59)(48,60), (1,44)(2,41)(3,42)(4,43)(5,22)(6,23)(7,24)(8,21)(9,32)(10,29)(11,30)(12,31)(13,18)(14,19)(15,20)(16,17)(25,61)(26,62)(27,63)(28,64)(33,56)(34,53)(35,54)(36,55)(37,60)(38,57)(39,58)(40,59)(45,52)(46,49)(47,50)(48,51), (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), (1,54,12,57)(2,53,9,60)(3,56,10,59)(4,55,11,58)(5,48,16,26)(6,47,13,25)(7,46,14,28)(8,45,15,27)(17,62,22,51)(18,61,23,50)(19,64,24,49)(20,63,21,52)(29,40,42,33)(30,39,43,36)(31,38,44,35)(32,37,41,34)>;
G:=Group( (1,15)(2,16)(3,13)(4,14)(5,9)(6,10)(7,11)(8,12)(17,41)(18,42)(19,43)(20,44)(21,31)(22,32)(23,29)(24,30)(25,56)(26,53)(27,54)(28,55)(33,61)(34,62)(35,63)(36,64)(37,51)(38,52)(39,49)(40,50)(45,57)(46,58)(47,59)(48,60), (1,44)(2,41)(3,42)(4,43)(5,22)(6,23)(7,24)(8,21)(9,32)(10,29)(11,30)(12,31)(13,18)(14,19)(15,20)(16,17)(25,61)(26,62)(27,63)(28,64)(33,56)(34,53)(35,54)(36,55)(37,60)(38,57)(39,58)(40,59)(45,52)(46,49)(47,50)(48,51), (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), (1,54,12,57)(2,53,9,60)(3,56,10,59)(4,55,11,58)(5,48,16,26)(6,47,13,25)(7,46,14,28)(8,45,15,27)(17,62,22,51)(18,61,23,50)(19,64,24,49)(20,63,21,52)(29,40,42,33)(30,39,43,36)(31,38,44,35)(32,37,41,34) );
G=PermutationGroup([[(1,15),(2,16),(3,13),(4,14),(5,9),(6,10),(7,11),(8,12),(17,41),(18,42),(19,43),(20,44),(21,31),(22,32),(23,29),(24,30),(25,56),(26,53),(27,54),(28,55),(33,61),(34,62),(35,63),(36,64),(37,51),(38,52),(39,49),(40,50),(45,57),(46,58),(47,59),(48,60)], [(1,44),(2,41),(3,42),(4,43),(5,22),(6,23),(7,24),(8,21),(9,32),(10,29),(11,30),(12,31),(13,18),(14,19),(15,20),(16,17),(25,61),(26,62),(27,63),(28,64),(33,56),(34,53),(35,54),(36,55),(37,60),(38,57),(39,58),(40,59),(45,52),(46,49),(47,50),(48,51)], [(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)], [(1,54,12,57),(2,53,9,60),(3,56,10,59),(4,55,11,58),(5,48,16,26),(6,47,13,25),(7,46,14,28),(8,45,15,27),(17,62,22,51),(18,61,23,50),(19,64,24,49),(20,63,21,52),(29,40,42,33),(30,39,43,36),(31,38,44,35),(32,37,41,34)]])
C22×C4⋊C4 is a maximal subgroup of
C24.625C23 C24.626C23 C24.5Q8 C24.631C23 C24.632C23 C24.634C23 C24.635C23 (C2×C4)⋊M4(2) C24.152D4 (C22×C4).275D4 C23.36D8 C24.157D4 C23.37D8 C24.159D4 C24.160D4 C23.38D8 C24.524C23 C24.542C23 C23.195C24 C24.545C23 C23.199C24 C24.195C23 C23.226C24 C23.227C24 C23.234C24 C24.558C23 C24.215C23 C24.243C23 C23.313C24 C23.316C24 C24.252C23 C24.568C23 C24.569C23 C24.269C23 C23.349C24 C24.572C23 C24.573C23 C24.576C23 C24.299C23 C24.300C23 C23.401C24 C23.402C24 C24.579C23 C23.404C24 C23.479C24 C23.483C24 C23.491C24 C24.587C23 C24.589C23 C23.527C24 C24.183D4 D4×C22×C4 Q8×C22×C4 C22.81C25
C22×C4⋊C4 is a maximal quotient of
C23.167C24 C23.178C24 C24.91D4 C24.545C23 C23.199C24 C23.231C24 C23.233C24 C42.257C23 C42.674C23 C24.100D4 C4○D4.7Q8 C4○D4.8Q8 M4(2).29C23
40 conjugacy classes
| class | 1 | 2A | ··· | 2O | 4A | ··· | 4X |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | - | |
| image | C1 | C2 | C2 | C4 | D4 | Q8 |
| kernel | C22×C4⋊C4 | C2×C4⋊C4 | C23×C4 | C22×C4 | C23 | C23 |
| # reps | 1 | 12 | 3 | 16 | 4 | 4 |
Matrix representation of C22×C4⋊C4 ►in GL5(𝔽5)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 4 | 0 |
| 2 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 2 | 0 |
G:=sub<GL(5,GF(5))| [1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,1,0],[2,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,2,0] >;
C22×C4⋊C4 in GAP, Magma, Sage, TeX
C_2^2\times C_4\rtimes C_4
% in TeX
G:=Group("C2^2xC4:C4"); // GroupNames label
G:=SmallGroup(64,194);
// by ID
G=gap.SmallGroup(64,194);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,192,217,103]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations